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Capture, Propagate, & Control Distributional Un...

Florian Dörfler
October 16, 2024
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Capture, Propagate, & Control Distributional Uncertainty

Nordic Congress of Mathematicians 2023

Florian Dörfler

October 16, 2024
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  1. Capture, Propagate, & Control Distributional Uncertainty Florian Dörfler & Liviu

    Aolaritei ETH Zürich Automatic Control Laboratory 1
  2. The Two Canonical Uncertainty Descriptions <latexit sha1_base64="bKk7/nYDpKrr5I+zVjxStZt6mSU=">AAACBXicbVDLSgMxFL1TX7W+qi51ESyCqzJTirosCOKygn1AO5RMJtOGJpkhyYildOPGX3HjQhG3/oM7/8a0nYW2HggczrmXm3OChDNtXPfbya2srq1v5DcLW9s7u3vF/YOmjlNFaIPEPFbtAGvKmaQNwwyn7URRLAJOW8Hwauq37qnSLJZ3ZpRQX+C+ZBEj2FipVzzu6ijCgvERumYPNEShvalYkM7tklt2Z0DLxMtICTLUe8WvbhiTVFBpCMdadzw3Mf4YK8MIp5NCN9U0wWSI+7RjqcSCan88SzFBp1YJURQr+6RBM/X3xhgLrUcisJMCm4Fe9Kbif14nNdGlP2YySQ2VZH4oSjkyMZpWYiMrSoxtIGSYKGb/isgAK0yMLa5gS/AWIy+TZqXsnZert5VSrZrVkYcjOIEz8OACanADdWgAgUd4hld4c56cF+fd+ZiP5pxs5xD+wPn8Adf3mMM=</latexit> Fixed distribution <latexit

    sha1_base64="kwScICG0h4iLhqpo8smL5yywEJM=">AAACBnicbZDLSgMxFIYz9VbrbdSlCMEiuCozpajLghuXFe0F2qFkMpk2NJkZkjNiGbpy46u4caGIW5/BnW9j2s5CW38I/PznnCTn8xPBNTjOt1VYWV1b3yhulra2d3b37P2Dlo5TRVmTxiJWHZ9oJnjEmsBBsE6iGJG+YG1/dDWtt++Z0jyO7mCcME+SQcRDTgmYqG8f93QYEsnFGPeAPYAfZrcQ0yHRwOmkb5edijMTXjZubsooV6Nvf/WCmKaSRUAF0brrOgl4GVHmNsEmpV6qWULoiAxY19iISKa9bLbGBJ+aJMBhrMyJAM/S3xMZkVqPpW86JYGhXqxNw/9q3RTCSy/jUZICi+j8oTAVGGI8ZYIDrhgFgyDghCpu/ooNAUUoGHIlA8FdXHnZtKoV97xSu6mW67UcRxEdoRN0hlx0geroGjVQE1H0iJ7RK3qznqwX6936mLcWrHzmEP2R9fkDOf2Zkw==</latexit> Stochastic <latexit sha1_base64="rxOYiltHKfJR27RnrbbCSuAqdps=">AAACAnicbVDLSsNAFJ3UV62vqCtxM1gEVyUpRV0W3LisYB/QhjKZ3LRDZ5IwMymWUNz4K25cKOLWr3Dn3zhts9DWAwOHc+7hzj1+wpnSjvNtFdbWNza3itulnd29/QP78Kil4lRSaNKYx7LjEwWcRdDUTHPoJBKI8Dm0/dHNzG+PQSoWR/d6koAnyCBiIaNEG6lvn/RUGBLB+ASLWAKGBxNXio2hb5edijMHXiVuTsooR6Nvf/WCmKYCIk05UarrOon2MiI1oxympV6qICF0RAbQNTQiApSXzU+Y4nOjBDiMpXmRxnP1dyIjQqmJ8M2kIHqolr2Z+J/XTXV47WUsSlINEV0sClOOdYxnfeCASaDanB8wQiUzf8V0SCSh2rRWMiW4yyevkla14l5WanfVcr2W11FEp+gMXSAXXaE6ukUN1EQUPaJn9IrerCfrxXq3PhajBSvPHKM/sD5/ALGYl5Y=</latexit> more expressive <latexit sha1_base64="tWPV48wV1scI2Piq93SDFGm2BFY=">AAACBHicbVC7SgNBFJ31GeNr1TLNYBCswm4IahmwsYxgHpAsYXb2bjJkdmaZmQ2EJYWNv2JjoYitH2Hn3zh5FJp44MLhnHu5954w5Uwbz/t2Nja3tnd2C3vF/YPDo2P35LSlZaYoNKnkUnVCooEzAU3DDIdOqoAkIYd2OLqd+e0xKM2keDCTFIKEDASLGSXGSn231NNxTBLGJ5iD1phKoUGNrTuGvlv2Kt4ceJ34S1JGSzT67lcvkjRLQBjKidZd30tNkBNlGOUwLfYyDSmhIzKArqWCJKCDfP7EFF9YJcKxVLaEwXP190ROEq0nSWg7E2KGetWbif953czEN0HORJoZEHSxKM44NhLPEsERU0CNDSBihCpmb8V0SBShxuZWtCH4qy+vk1a14l9VavfVcr22jKOASugcXSIfXaM6ukMN1EQUPaJn9IrenCfnxXl3PhatG85y5gz9gfP5A0nPmHk=</latexit> less conservative <latexit sha1_base64="vVSFQJImOjAhu8hgKmYB6epM/DQ=">AAACBHicbVDLSsNAFJ3UV62vqMtuBovgqiSlqMuCG5cV7APaUG4mk3boTBJmJkIJXbjxV9y4UMStH+HOv3HSZqGtBwYO59zLnXP8hDOlHefbKm1sbm3vlHcre/sHh0f28UlXxakktENiHsu+D4pyFtGOZprTfiIpCJ/Tnj+9yf3eA5WKxdG9niXUEzCOWMgIaCON7OpQhSEIxmd4AjLAOsaJjBMYg6Yju+bUnQXwOnELUkMF2iP7axjEJBU00oSDUgPXSbSXgdSMcDqvDFNFEyBTGNOBoREIqrxsEWKOz40S4DCW5kUaL9TfGxkIpWbCN5MC9EStern4nzdIdXjtZSxKUk0jsjwUpjyPmjeCAyYp0aaAgAGRzPwVE9MFEG16q5gS3NXI66TbqLuX9eZdo9ZqFnWUURWdoQvkoivUQreojTqIoEf0jF7Rm/VkvVjv1sdytGQVO6foD6zPH6QemA4=</latexit> hard to propagate <latexit sha1_base64="NCanRF85KT/lNkJI29E7p9UmDLo=">AAACAXicbVDLSsNAFJ3UV62vqBvBzWARXJWkFHVZcOOygn1AG8rNZNIOnUnCzEQooW78FTcuFHHrX7jzb5y0WWjrgYHDOfdw5x4/4Uxpx/m2SmvrG5tb5e3Kzu7e/oF9eNRRcSoJbZOYx7Lng6KcRbStmea0l0gKwue0609ucr/7QKVicXSvpwn1BIwiFjIC2khD+2SgwhAE41PMqVJYSyAaTHpoV52aMwdeJW5BqqhAa2h/DYKYpIJGmnBQqu86ifYykJoRTmeVQapoAmQCI9o3NAJBlZfNL5jhc6MEOIyleZHGc/V3IgOh1FT4ZlKAHqtlLxf/8/qpDq+9jEVJqmlEFovClGMd47wOHDBJiTbXBwyIZOavmIwhL8GUVjEluMsnr5JOveZe1hp39WqzUdRRRqfoDF0gF12hJrpFLdRGBD2iZ/SK3qwn68V6tz4WoyWryByjP7A+fwCcB5b0</latexit> less tractable <latexit sha1_base64="k1AtdIimUWBqygfvFGKKOZEHPFk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE8lUSKeix48VjBfkAaymY7aZdudsPupFBCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXpYIb9LxvZ2Nza3tnt7RX3j84PDqunJy2jco0gxZTQuluRA0ILqGFHAV0Uw00iQR0ovH93O9MQBuu5BNOUwgTOpQ85oyilYLehGpIDRdK9itVr+Yt4K4TvyBVUqDZr3z1BoplCUhkghoT+F6KYU41ciZgVu5lBlLKxnQIgaWSJmDCfHHyzL20ysCNlbYl0V2ovydymhgzTSLbmVAcmVVvLv7nBRnGd2HOZZohSLZcFGfCReXO/3cHXANDMbWEMs3trS4bUU0Z2pTKNgR/9eV10r6u+Te1+mO92qgXcZTIObkgV8Qnt6RBHkiTtAgjijyTV/LmoPPivDsfy9YNp5g5I3/gfP4AuIeRgg==</latexit> " <latexit sha1_base64="80NlNJstWKQugZ8ql5cDaQ9yTM4=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWAR3FiSUtRlwY0rqWAf0IYymUzaofMIMxOhxC78FTcuFHHrb7jzb5y2WWjrgQuHc+7l3nvChFFtPO/bKaysrq1vFDdLW9s7u3vu/kFLy1Rh0sSSSdUJkSaMCtI01DDSSRRBPGSkHY6up377gShNpbg344QEHA0EjSlGxkp996in4xhxysbwVip+HspURCTqu2Wv4s0Al4mfkzLI0ei7X71I4pQTYTBDWnd9LzFBhpShmJFJqZdqkiA8QgPStVQgTnSQze6fwFOrRDCWypYwcKb+nsgQ13rMQ9vJkRnqRW8q/ud1UxNfBRkVSWqIwPNFccqgkXAaBoyoItjY3yOKsKL2VoiHSCFsbGQlG4K/+PIyaVUr/kWldlct12t5HEVwDE7AGfDBJaiDG9AATYDBI3gGr+DNeXJenHfnY95acPKZQ/AHzucP7R2WAQ==</latexit> Norm-bounded <latexit sha1_base64="d1/x9BXeKVV3B47JfmQ9BZTKshA=">AAAB/nicbVBNS8NAEN34WetXVDx5WSyCp5KUoh4LXjxWsB/QhrLZbtqlu5uwOxFCKPhXvHhQxKu/w5v/xm2bg7Y+GHi8N8PMvDAR3IDnfTtr6xubW9ulnfLu3v7BoXt03DZxqilr0VjEuhsSwwRXrAUcBOsmmhEZCtYJJ7czv/PItOGxeoAsYYEkI8UjTglYaeCe9k0UEclFhlNFmQbCFWQDt+JVvTnwKvELUkEFmgP3qz+MaSqZAiqIMT3fSyDIiQZOBZuW+6lhCaETMmI9SxWRzAT5/PwpvrDKEEextqUAz9XfEzmRxmQytJ2SwNgsezPxP6+XQnQT5FwlKTBFF4uiVGCI8SwLPOSaUbCvDzmhmttbMR0TTSjYxMo2BH/55VXSrlX9q2r9vlZp1Is4SugMnaNL5KNr1EB3qIlaiKIcPaNX9OY8OS/Ou/OxaF1zipkT9AfO5w/UfpYD</latexit> uncertainty <latexit sha1_base64="8y3vxxSx5bFIxRKa48MXiGkTBDA=">AAACAnicbVDLSsNAFJ3UV62vqCtxM1gEVyUpRV0W3LisYh/QhjKZTNqhM0mYuRFLKG78FTcuFHHrV7jzb5y2WWjrgYHDOfdy5xw/EVyD43xbhZXVtfWN4mZpa3tnd8/eP2jpOFWUNWksYtXxiWaCR6wJHATrJIoR6QvW9kdXU799z5TmcXQH44R5kgwiHnJKwEh9+6inw5BILsa4B+wB/DC7jf1Uw6Rvl52KMwNeJm5OyihHo29/9YKYppJFQAXRuus6CXgZUcCpYJNSL9UsIXREBqxraEQk0142izDBp0YJcBgr8yLAM/X3Rkak1mPpm0lJYKgXvan4n9dNIbz0Mh4lKbCIzg+FqcAQ42kfOOCKUTDxA06o4uavmA6JIhRMayVTgrsYeZm0qhX3vFK7qZbrtbyOIjpGJ+gMuegC1dE1aqAmougRPaNX9GY9WS/Wu/UxHy1Y+c4h+gPr8wcS25fV</latexit> Robust <latexit sha1_base64="pr8wHeVxQt6QaCSEUqCsPN0QM0M=">AAACBnicbVDLSgMxFM3UV62vUZciBIvgqsyUoi4LblxWsA9oh5JJ77ShmcyQZIrD0JUbf8WNC0Xc+g3u/BvTdhbaeiBwOOfcJPf4MWdKO863VVhb39jcKm6Xdnb39g/sw6OWihJJoUkjHsmOTxRwJqCpmebQiSWQ0OfQ9sc3M789AalYJO51GoMXkqFgAaNEG6lvn/ZUEJCQ8RSLSGMTTTE8mCuUYhPo22Wn4syBV4mbkzLK0ejbX71BRJMQhKacKNV1nVh7GZGaUQ7TUi9REBM6JkPoGipICMrL5mtM8blRBjiIpDlC47n6eyIjoVJp6JtkSPRILXsz8T+vm+jg2suYiBMNgi4eChKOdYRnneABk0C1qWDACJXM/BXTEZGEatNcyZTgLq+8SlrVintZqd1Vy/VaXkcRnaAzdIFcdIXq6BY1UBNR9Iie0St6s56sF+vd+lhEC1Y+c4z+wPr8AbscmUI=</latexit> not very expressive <latexit sha1_base64="cLrvjTWQrmHzPmDFAy+G+FH0dSc=">AAAB/3icbVA9SwNBEN3zM8avU8HGZjEIVuEuBLUM2FhGMB+QHGFvM5cs2d07dvcC4UzhX7GxUMTWv2Hnv3GTXKGJDwYe780wMy9MONPG876dtfWNza3twk5xd2//4NA9Om7qOFUUGjTmsWqHRANnEhqGGQ7tRAERIYdWOLqd+a0xKM1i+WAmCQSCDCSLGCXGSj33tKujiAjGJ5jGUoMaW2MMPbfklb058Crxc1JCOeo996vbj2kqQBrKidYd30tMkBFlGOUwLXZTDQmhIzKAjqWSCNBBNr9/ii+s0sdRrGxJg+fq74mMCK0nIrSdgpihXvZm4n9eJzXRTZAxmaQGJF0silKOTYxnYeA+U0CN/b3PCFXM3orpkChCjY2saEPwl19eJc1K2b8qV+8rpVo1j6OAztA5ukQ+ukY1dIfqqIEoekTP6BW9OU/Oi/PufCxa15x85gT9gfP5A5hDlnA=</latexit> conservative <latexit sha1_base64="/qCZW2bqqU1GCODuHYQC5cKD1dk=">AAACBHicbVC7SgNBFJ2Nrxhfq5ZpBoNgFXZDUMuAjWUE84AkhLuTu8mQmd1lZlZYQgobf8XGQhFbP8LOv3HyKDTxwMDhnHu5c06QCK6N5307uY3Nre2d/G5hb//g8Mg9PmnqOFUMGywWsWoHoFHwCBuGG4HtRCHIQGArGN/M/NYDKs3j6N5kCfYkDCMecgbGSn232NVhCJKLjCLojJqYJipOYAgG+27JK3tz0HXiL0mJLFHvu1/dQcxSiZFhArTu+F5iehNQhjOB00I31ZgAG8MQO5ZGIFH3JvMQU3pulQENY2VfZOhc/b0xAal1JgM7KcGM9Ko3E//zOqkJr3sTHiWpwYgtDoWpmEWdNUIHXCEztoABB6a4/StlI1DAjO2tYEvwVyOvk2al7F+Wq3eVUq26rCNPiuSMXBCfXJEauSV10iCMPJJn8krenCfnxXl3PhajOWe5c0r+wPn8AcHZmCE=</latexit> easy to propagate <latexit sha1_base64="TEQCUnm1+i6nfbeLMVyNxvBrWg8=">AAAB/HicbVBNS8NAFNzUr1q/oj16WSyCp5KUoh4LXjxWsK3QhrLZbNqlu0nYfRFCqH/FiwdFvPpDvPlv3LQ5aOvAwjDzhn1v/ERwDY7zbVU2Nre2d6q7tb39g8Mj+/ikr+NUUdajsYjVg080EzxiPeAg2EOiGJG+YAN/dlP4g0emNI+je8gS5kkyiXjIKQEjje36SIchkVxkGBShQExwbDecprMAXiduSRqoRHdsf42CmKaSRUAF0XroOgl4OVHAqWDz2ijVLCF0RiZsaGhEJNNevlh+js+NEuAwVuZFgBfq70ROpNaZ9M2kJDDVq14h/ucNUwivvZxHSQososuPwlRgiHHRBA64YhTM4QEnVHGzK6ZTUpRg+qqZEtzVk9dJv9V0L5vtu1aj0y7rqKJTdIYukIuuUAfdoi7qIYoy9Ixe0Zv1ZL1Y79bHcrRilZk6+gPr8wfwHpTr</latexit> tractable 7 Why Is Forecasting Crucial to Integrating Variable RE to the Grid? Source: Pierre Pinson, DTU, Denmark Average day ahead error: 8%-10% for wind farm, 4% for system Ramp error: Over 50% for large ramps Operating with increased uncertainty… <latexit sha1_base64="mbACyP+RcTCnxOUVDOtc9wr3AuA=">AAAB+3icbVBNS8NAEN3Ur1q/Yj16WSyCp5KUoh4LXjxWsK3QhjLZbNqlu5uwuxFL6F/x4kERr/4Rb/4bt20O2vpg4PHeDDPzwpQzbTzv2yltbG5t75R3K3v7B4dH7nG1q5NMEdohCU/UQwiaciZpxzDD6UOqKIiQ0144uZn7vUeqNEvkvZmmNBAwkixmBIyVhm51oOMYBONTbBQwyeRo6Na8urcAXid+QWqoQHvofg2ihGSCSkM4aN33vdQEOSjDCKezyiDTNAUygRHtWypBUB3ki9tn+NwqEY4TZUsavFB/T+QgtJ6K0HYKMGO96s3F/7x+ZuLrIGcyzQyVZLkozjg2CZ4HgSOmKDH274gBUczeiskYFBBj46rYEPzVl9dJt1H3L+vNu0at1SziKKNTdIYukI+uUAvdojbqIIKe0DN6RW/OzHlx3p2PZWvJKWZO0B84nz9Bn5SL</latexit> training <latexit sha1_base64="e4Wijgi2KUeRpvMclYsaAss3fAc=">AAAB+nicbVBNSwMxEM3Wr1q/tnr0EiyCp7JbinosePFYwX5Au5Rsmm1Dk+ySzCpl7U/x4kERr/4Sb/4b03YP2vpg4PHeDDPzwkRwA5737RQ2Nre2d4q7pb39g8Mjt3zcNnGqKWvRWMS6GxLDBFesBRwE6yaaERkK1gknN3O/88C04bG6h2nCAklGikecErDSwC33TRQRycUUAzPA1WjgVryqtwBeJ35OKihHc+B+9YcxTSVTQAUxpud7CQQZ0cCpYLNSPzUsIXRCRqxnqSKSmSBbnD7D51YZ4ijWthTghfp7IiPSmKkMbackMDar3lz8z+ulEF0HGVdJCkzR5aIoFRhiPM8BD7lmFOzbQ06o5vZWTMdEEwo2rZINwV99eZ20a1X/slq/q1Ua9TyOIjpFZ+gC+egKNdAtaqIWougRPaNX9OY8OS/Ou/OxbC04+cwJ+gPn8weHTJQj</latexit> testing additionally: both require prior knowledge & suffer from distribution shift … can we do better ? 2 distribution 
 shift
  3. Distributional Uncertainty via Optimal Transport <latexit sha1_base64="/+FRpYVTL/Q5DmLt2ZGqfvDazEI=">AAACAnicbVDLSsNAFL3xWesr6krcDBbBVUmkqMuCG5cV7AOaUCbTaTt0MgkzEyGE4sZfceNCEbd+hTv/xkmahbYeGDicc19zgpgzpR3n21pZXVvf2KxsVbd3dvf27YPDjooSSWibRDySvQArypmgbc00p71YUhwGnHaD6U3udx+oVCwS9zqNqR/isWAjRrA20sA+zrxiSEZSLGZeiPUkCFBrIGYDu+bUnQJombglqUGJ1sD+8oYRSUIqNOFYqb7rxNrPsNSMcDqreomiMSZTPKZ9QwUOqfKzYvsMnRlliEaRNE9oVKi/OzIcKpWGganMb1SLXi7+5/UTPbr2MybiRFNB5otGCUc6QnkeaMgkJZqnhmAimbkVkQmWmGiTWtWE4C5+eZl0LuruZb1x16g1G2UcFTiBUzgHF66gCbfQgjYQeIRneIU368l6sd6tj3npilX2HMEfWJ8/8cqXvQ==</latexit> P n <latexit

    sha1_base64="k1AtdIimUWBqygfvFGKKOZEHPFk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE8lUSKeix48VjBfkAaymY7aZdudsPupFBCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXpYIb9LxvZ2Nza3tnt7RX3j84PDqunJy2jco0gxZTQuluRA0ILqGFHAV0Uw00iQR0ovH93O9MQBuu5BNOUwgTOpQ85oyilYLehGpIDRdK9itVr+Yt4K4TvyBVUqDZr3z1BoplCUhkghoT+F6KYU41ciZgVu5lBlLKxnQIgaWSJmDCfHHyzL20ysCNlbYl0V2ovydymhgzTSLbmVAcmVVvLv7nBRnGd2HOZZohSLZcFGfCReXO/3cHXANDMbWEMs3trS4bUU0Z2pTKNgR/9eV10r6u+Te1+mO92qgXcZTIObkgV8Qnt6RBHkiTtAgjijyTV/LmoPPivDsfy9YNp5g5I3/gfP4AuIeRgg==</latexit> " <latexit sha1_base64="gzBBd/K4jjBYVFimyExo36VKlsI=">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</latexit> P n = 1 n n X i=1 zi <latexit sha1_base64="sjlPuQ12qw6GUmCDlxYS0CfJcIU=">AAAB/HicbVDLSsNAFL3xWesr2qWbwSK4KokUdVlw47KCfUAbwmQ6bYdOJmFmIsQQf8WNC0Xc+iHu/BunaRbaeuDC4Zx75849QcyZ0o7zba2tb2xubVd2qrt7+weH9tFxV0WJJLRDIh7JfoAV5UzQjmaa034sKQ4DTnvB7Gbu9x6oVCwS9zqNqRfiiWBjRrA2km/XsmxYvJKRFIv80Xfz3LfrTsMpgFaJW5I6lGj79tdwFJEkpEITjpUauE6svQxLzQineXWYKBpjMsMTOjBU4JAqLyvW5ujMKCM0jqQpoVGh/p7IcKhUGgamM8R6qpa9ufifN0j0+NrLmIgTTQVZLBonHOkIzZNAIyYp0Tw1BBPJzF8RmWKJiTZ5VU0I7vLJq6R70XAvG827Zr3VLOOowAmcwjm4cAUtuIU2dIBACs/wCm/Wk/VivVsfi9Y1q5ypwR9Ynz/PZJV8</latexit> z1 <latexit 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sha1_base64="GUD9jfdS/NnObjoPykcSdja/fbk=">AAACIHicbVDNS8MwHE39nPNr6tFLcAjzMloZTgRh6MXjBPcBay1plm1haVqSdFBK/xQv/itePCiiN/1rTLcKuvkg8Hjv/ZJfnhcyKpVpfhpLyyura+uFjeLm1vbObmlvvy2DSGDSwgELRNdDkjDKSUtRxUg3FAT5HiMdb3yd+Z0JEZIG/E7FIXF8NOR0QDFSWnJLddtHauR58MpN7AkSJJSUBTy9x5XEnl6f4Bjx9CfWdHl6Ai8u3VLZrJpTwEVi5aQMcjTd0ofdD3DkE64wQ1L2LDNUToKEopiRtGhHkoQIj9GQ9DTlyCfSSaYbpPBYK304CIQ+XMGp+nsiQb6Use/pZLannPcy8T+vF6nBuZNQHkaKcDx7aBAxqAKYtQX7VBCsWKwJwoLqXSEeIYGw0p0WdQnW/JcXSfu0ap1Va7e1cqOW11EAh+AIVIAF6qABbkATtAAGD+AJvIBX49F4Nt6M91l0ychnDsAfGF/fd8mjtA==</latexit> Bc " (P n) := <latexit sha1_base64="pwdxrW+hpeJMkJ16Y3YgW/5to9I=">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</latexit> OT distance Wc(Q, P) can capture continuous (physics) & discrete (data) distributions ! data-driven parameterization 3 <latexit sha1_base64="kX4N/i38TmnWl3Cc1tEqlX2dBUI=">AAACBnicbVDLSgNBEJyNrxhfqx5FGAyCp7AbgnoMePEYwTwgWcLs7GwyZHZmmekVwpKTF3/FiwdFvPoN3vwbJ4+DJhY0FFXddHeFqeAGPO/bKaytb2xuFbdLO7t7+wfu4VHLqExT1qRKKN0JiWGCS9YEDoJ1Us1IEgrWDkc3U7/9wLThSt7DOGVBQgaSx5wSsFLfPe2ZOCYJF2MMmkiTKg0zC1NloO+WvYo3A14l/oKU0QKNvvvVixTNEiaBCmJM1/dSCHKigVPBJqVeZlhK6IgMWNdSSRJmgnz2xgSfWyXCsdK2JOCZ+nsiJ4kx4yS0nQmBoVn2puJ/XjeD+DrIuUwzYJLOF8WZwKDwNBMccc0o2AgiTqjm9lZMh0QTCja5kg3BX355lbSqFf+yUrurluu1RRxFdILO0AXy0RWqo1vUQE1E0SN6Rq/ozXlyXpx352PeWnAWM8foD5zPHxuLmX4=</latexit> transportation cost <latexit 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sha1_base64="i5sLMMt1kFhTMAZpKmgweGRKmoo=">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</latexit> with marginals Q and P <latexit sha1_base64="9eCoAcYguvMh78nYBFBSGarU0M0=">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</latexit> z <latexit sha1_base64="PzJ5v5XBA5/BxHrgzxXfo53fGHc=">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</latexit> x <latexit sha1_base64="ssKGA8k9tWBg01JUBYBjjEs1m/s=">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</latexit> Wc(Q, P) = inf 2 (Q,P) E (x,z)⇠ [ c(x z) ] <latexit sha1_base64="SXsVChNVuGEdSRjFh944XrzGyhY=">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</latexit> (x, z) <latexit sha1_base64="ssKGA8k9tWBg01JUBYBjjEs1m/s=">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</latexit> Wc(Q, P) = inf 2 (Q,P) E (x,z)⇠ [ c(x z) ] <latexit sha1_base64="JesJOUGqTIO1TJtO2Kawry+Pwps=">AAACB3icbVC7SgNBFJ31GeMrainIYBCswm4IahmwsTNCXpCEMDu5mwyZnV1m7ophSWfjr9hYKGLrL9j5N04ehSYeGDiccy93zvFjKQy67rezsrq2vrGZ2cpu7+zu7ecODusmSjSHGo9kpJs+MyCFghoKlNCMNbDQl9Dwh9cTv3EP2ohIVXEUQydkfSUCwRlaqZs7aZsgYKGQI9pGeEA/SG+rtGcPM8Vh3M3l3YI7BV0m3pzkyRyVbu6r3Yt4EoJCLpkxLc+NsZMyjYJLGGfbiYGY8SHrQ8tSxUIwnXSaY0zPrNKjQaTtU0in6u+NlIXGjELfToYMB2bRm4j/ea0Eg6tOKlScICg+OxQkkmJEJ6XYvBo42g56gnEt7F8pHzDNONrqsrYEbzHyMqkXC95FoXRXzJdL8zoy5JicknPikUtSJjekQmqEk0fyTF7Jm/PkvDjvzsdsdMWZ7xyRP3A+fwBcYZmW</latexit> OT distance <latexit sha1_base64="ssKGA8k9tWBg01JUBYBjjEs1m/s=">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</latexit> Wc(Q, P) = inf 2 (Q,P) E (x,z)⇠ [ c(x z) ] <latexit sha1_base64="ssKGA8k9tWBg01JUBYBjjEs1m/s=">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</latexit> Wc(Q, P) = inf 2 (Q,P) E (x,z)⇠ [ c(x z) ]
  4. Geometric Properties of OT Ambiguity Sets <latexit sha1_base64="QuFmWrDOFQ3u9y9HyOjlIVEhDq4=">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</latexit> Example. P

    1 = 0 <latexit sha1_base64="k1AtdIimUWBqygfvFGKKOZEHPFk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE8lUSKeix48VjBfkAaymY7aZdudsPupFBCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXpYIb9LxvZ2Nza3tnt7RX3j84PDqunJy2jco0gxZTQuluRA0ILqGFHAV0Uw00iQR0ovH93O9MQBuu5BNOUwgTOpQ85oyilYLehGpIDRdK9itVr+Yt4K4TvyBVUqDZr3z1BoplCUhkghoT+F6KYU41ciZgVu5lBlLKxnQIgaWSJmDCfHHyzL20ysCNlbYl0V2ovydymhgzTSLbmVAcmVVvLv7nBRnGd2HOZZohSLZcFGfCReXO/3cHXANDMbWEMs3trS4bUU0Z2pTKNgR/9eV10r6u+Te1+mO92qgXcZTIObkgV8Qnt6RBHkiTtAgjijyTV/LmoPPivDsfy9YNp5g5I3/gfP4AuIeRgg==</latexit> " <latexit sha1_base64="xmF6pZFY52xDJuqZaH5ptjhsUXM=">AAACGHicbZDLSgMxFIYz9V5vVZdugkVwVWdKUZcFF7pUsBdoS8lkzrShSWbIyQi19DHc+CpuXCji1p1vY1q70NYfAj//OSfJ+cJUCrS+/+XllpZXVtfWN/KbW9s7u4W9/TommeFQ44lMTDNkCFJoqFlhJTRTA0yFEhrh4HJSb9yDQZHoOztMoaNYT4tYcGZd1C2ctjGOmRJySK9Ag2FSPADSCCwYJbT7gODUJGGGVgMi7RaKfsmfii6aYGaKZKabbuGzHSU8U6AtlwyxFfip7YyYcRdLGOfbGULK+ID1oOWsZgqwM5ouNqbHLolonBh3tKXT9PfEiCnEoQpdp2K2j/O1SfhfrZXZ+KIzEjrNLGj+81CcSWoTOqFEI2GAWwclEowbMYHA+8ww7rBg3kEI5ldeNPVyKTgrVW7LxWplhmOdHJIjckICck6q5JrckBrh5JE8k1fy5j15L9679/HTmvNmMwfkj7zPbwFjoPY=</latexit> Generalizes deterministic robustness <latexit sha1_base64="sLqJOk8XThAT97sixyNrT6RKHjQ=">AAACAnicbVC7SgNBFJ2Nrxhfq1ZiMxgEq7AbgloGbCwjmAckS5idnU2GzGOZhxBCsPFXbCwUsfUr7PwbJ8kWmnhg4HDOvdw5J84Y1SYIvr3C2vrG5lZxu7Szu7d/4B8etbS0CpMmlkyqTow0YVSQpqGGkU6mCOIxI+14dDPz2w9EaSrFvRlnJOJoIGhKMTJO6vsnPZ2miFM2hrG0IiEJ1DbLpDJ9vxxUgjngKglzUgY5Gn3/q5dIbDkRBjOkdTcMMhNNkDIUMzIt9awmGcIjNCBdRwXiREeTeYQpPHdKAlOp3BMGztXfGxPEtR7z2E1yZIZ62ZuJ/3lda9LraEJFZg0ReHEotQwaCWd9wIQqgo2Ln1CEFXV/hXiIFMLGtVZyJYTLkVdJq1oJLyu1u2q5XsvrKIJTcAYuQAiuQB3cggZoAgwewTN4BW/ek/fivXsfi9GCl+8cgz/wPn8ArRqXkw==</latexit> bounded support <latexit sha1_base64="ZXn69+wzQl7ByKAiDd5rFoFKgyI=">AAACBHicbVC7SgNBFJ2Nrxhfq5ZpBoNgFXZDUMuAjWUE84BkCbOzs8mQeSzzEMKSwsZfsbFQxNaPsPNvnDwKTTxw4XDOvdx7T5wxqk0QfHuFjc2t7Z3ibmlv/+DwyD8+aWtpFSYtLJlU3RhpwqggLUMNI91MEcRjRjrx+Gbmdx6I0lSKezPJSMTRUNCUYmScNPDLfZ2miFM2gVbE0oqEJFDbLJPKDPxKUA3mgOskXJIKWKI58L/6icSWE2EwQ1r3wiAzUY6UoZiRaalvNckQHqMh6TkqECc6yudPTOG5UxKYSuVKGDhXf0/kiGs94bHr5MiM9Ko3E//zetak11FORWYNEXixKLUMGglnicCEKoKNCyChCCvqboV4hBTCxuVWciGEqy+vk3atGl5W63e1SqO+jKMIyuAMXIAQXIEGuAVN0AIYPIJn8ArevCfvxXv3PhatBW85cwr+wPv8AWUcmIo=</latexit> unbounded support <latexit sha1_base64="xlzEn4ZUDzFRkf4KinRRwhrs8Z4=">AAACEHicbVA9SwNBEN3z2/gVtbRZDKJVuJOgloKNZQQTA0kIc5u5uLi3d+zOqeHIT7Dxr9hYKGJraee/cRNTaOKDgcd7M7szL0yVtOT7X97M7Nz8wuLScmFldW19o7i5VbdJZgTWRKIS0wjBopIaayRJYSM1CHGo8Cq8ORv6V7dorEz0JfVTbMfQ0zKSAshJneJ+y0YRxFL1uYCUMoOWtwjvKYzyUIG44fYOtB10iiW/7I/Ap0kwJiU2RrVT/Gx1E5HFqEkosLYZ+Cm1czAkhcJBoZVZTN370MOmoxpitO18dNCA7zmly6PEuNLER+rviRxia/tx6DpjoGs76Q3F/7xmRtFJO5c6zQi1+PkoyhSnhA/T4V1pUJALoytBGOl25eIaDAhyGRZcCMHkydOkflgOjsqVi8PSaWUcxxLbYbvsgAXsmJ2yc1ZlNSbYA3tiL+zVe/SevTfv/ad1xhvPbLM/8D6+AdjSnbI=</latexit> captures black swans <latexit sha1_base64="3Ohq6kucu8Z8VOc6F/5WZ7df1nw=">AAACC3icbVDLSgMxFM34rPU16tJNaBFclZlS1GXBjcsK9gHtUDKZO21oJhmSTKEM3bvxV9y4UMStP+DOvzF9LLT1QOBwzrnc3BOmnGnjed/OxubW9s5uYa+4f3B4dOyenLa0zBSFJpVcqk5INHAmoGmY4dBJFZAk5NAOR7czvz0GpZkUD2aSQpCQgWAxo8RYqe+WejqOScL4BCuIMgoaUyk0qLENjEGA1n237FW8OfA68ZekjJZo9N2vXiRploAwlBOtu76XmiAnyjDKYVrsZRpSQkdkAF1LBUlAB/n8lim+sEqEY6nsEwbP1d8TOUm0niShTSbEDPWqNxP/87qZiW+CnIk0MyDoYlGccWwknhWDI6aAGttDxAhVzP4V0yFRhBpbX9GW4K+evE5a1Yp/VandV8v12rKOAjpHJXSJfHSN6ugONVATUfSIntErenOenBfn3flYRDec5cwZ+gPn8wcR4pus</latexit> reduces conservativeness <latexit sha1_base64="nWp7s+XjDvWCQZglBO9+MeWv9Uw=">AAAB+nicbVDLSsNAFL2pr1pfqS7dBIvgqiRS1GXBjcsK9gFtCJPppB06mYSZiRJjPsWNC0Xc+iXu/BunaRbaeuDC4Zx75849fsyoVLb9bVTW1jc2t6rbtZ3dvf0Ds37Yk1EiMOniiEVi4CNJGOWkq6hiZBALgkKfkb4/u577/XsiJI34nUpj4oZowmlAMVJa8sx6NioeyXCKeP7oOblnNuymXcBaJU5JGlCi45lfo3GEk5BwhRmScujYsXIzJBTFjOS1USJJjPAMTchQU45CIt2s2Jpbp1oZW0EkdHFlFerviQyFUqahrztDpKZy2ZuL/3nDRAVXbkZ5nCjC8WJRkDBLRdY8B2tMBcGKpZogLKj+q4WnSCCsdFo1HYKzfPIq6Z03nYtm67bVaLfKOKpwDCdwBg5cQhtuoANdwPAAz/AKb8aT8WK8Gx+L1opRzhzBHxifP/1YlHA=</latexit> z1 <latexit sha1_base64="3RoHHu675qYqLgNkuwEbW/QbGPA=">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</latexit> ) 4 expressive: robustness in space & likelihood <latexit sha1_base64="ewySohb7cgEbB0upHtwIysKQQps=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Kkkp6rHgxWML9gPaUDbbSbt2swm7G6GE/gIvHhTx6k/y5r9x2+agrQ8GHu/NMDMvSATXxnW/nY3Nre2d3cJecf/g8Oi4dHLa1nGqGLZYLGLVDahGwSW2DDcCu4lCGgUCO8Hkbu53nlBpHssHM03Qj+hI8pAzaqzUdAelsltxFyDrxMtJGXI0BqWv/jBmaYTSMEG17nluYvyMKsOZwFmxn2pMKJvQEfYslTRC7WeLQ2fk0ipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDWz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m6INwVt9eZ20qxXvulJrVsv1Wh5HAc7hAq7Agxuowz00oAUMEJ7hFd6cR+fFeXc+lq0bTj5zBn/gfP4AdzGMrQ==</latexit> 0 <latexit sha1_base64="k1AtdIimUWBqygfvFGKKOZEHPFk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE8lUSKeix48VjBfkAaymY7aZdudsPupFBCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXpYIb9LxvZ2Nza3tnt7RX3j84PDqunJy2jco0gxZTQuluRA0ILqGFHAV0Uw00iQR0ovH93O9MQBuu5BNOUwgTOpQ85oyilYLehGpIDRdK9itVr+Yt4K4TvyBVUqDZr3z1BoplCUhkghoT+F6KYU41ciZgVu5lBlLKxnQIgaWSJmDCfHHyzL20ysCNlbYl0V2ovydymhgzTSLbmVAcmVVvLv7nBRnGd2HOZZohSLZcFGfCReXO/3cHXANDMbWEMs3trS4bUU0Z2pTKNgR/9eV10r6u+Te1+mO92qgXcZTIObkgV8Qnt6RBHkiTtAgjijyTV/LmoPPivDsfy9YNp5g5I3/gfP4AuIeRgg==</latexit> " <latexit sha1_base64="CT88sq2YM8AU5WkXJiKOQjZuGwA=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lKUY8FLx4r2FZoQ9lsJ+3SzSbubgol9Hd48aCIV3+MN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJW8epYthisYjVY0A1Ci6xZbgR+JgopFEgsBOMb+d+Z4JK81g+mGmCfkSHkoecUWMlv0Z6E6ow0VzEsl+uuFV3AbJOvJxUIEezX/7qDWKWRigNE1Trrucmxs+oMpwJnJV6qcaEsjEdYtdSSSPUfrY4ekYurDIgYaxsSUMW6u+JjEZaT6PAdkbUjPSqNxf/87qpCW/8jMskNSjZclGYCmJiMk+ADLhCZsTUEsoUt7cSNqKKMmNzKtkQvNWX10m7VvWuqvX7eqVRz+MowhmcwyV4cA0NuIMmtIDBEzzDK7w5E+fFeXc+lq0FJ585hT9wPn8AgoiR6A==</latexit> 2" <latexit sha1_base64="wi5TWe8g0FbVaEP0bKa26C1fuDw=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0qMeCF48VbC20oWy2k3bpZhN3N4US+ju8eFDEqz/Gm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCopeNUMWyyWMSqHVCNgktsGm4EthOFNAoEPgaj25n/OEaleSwfzCRBP6IDyUPOqLGSf0m6Y6ow0VzEsleuuFV3DrJKvJxUIEejV/7q9mOWRigNE1Trjucmxs+oMpwJnJa6qcaEshEdYMdSSSPUfjY/ekrOrNInYaxsSUPm6u+JjEZaT6LAdkbUDPWyNxP/8zqpCW/8jMskNSjZYlGYCmJiMkuA9LlCZsTEEsoUt7cSNqSKMmNzKtkQvOWXV0nroupdVWv3tUq9lsdRhBM4hXPw4BrqcAcNaAKDJ3iGV3hzxs6L8+58LFoLTj5zDH/gfP4AhBiR6Q==</latexit> 3" <latexit sha1_base64="uXKMM5aIxQypveRJvCOybOsJoq0=">AAAB83icbVBNSwMxEM3Wr1q/qh69BIvgxbIrRT0WvHisYD+gu5RsOtuGZpOQZAul9G948aCIV/+MN/+NabsHbX0w8Hhvhpl5seLMWN//9gobm1vbO8Xd0t7+weFR+fikZWSmKTSp5FJ3YmKAMwFNyyyHjtJA0phDOx7dz/32GLRhUjzZiYIoJQPBEkaJdVJ4FY6JBmUYl6JXrvhVfwG8ToKcVFCORq/8FfYlzVIQlnJiTDfwlY2mRFtGOcxKYWZAEToiA+g6KkgKJpoubp7hC6f0cSK1K2HxQv09MSWpMZM0dp0psUOz6s3F/7xuZpO7aMqEyiwIulyUZBxbiecB4D7TQC2fOEKoZu5WTIdEE2pdTCUXQrD68jppXVeDm2rtsVap1/I4iugMnaNLFKBbVEcPqIGaiCKFntErevMy78V79z6WrQUvnzlFf+B9/gAjsZG5</latexit> " <latexit sha1_base64="sbyI3f0EeArzzLyAfPSEEPXZngw=">AAAB9HicbVDLSgNBEOyNrxhfUY9eFoPgxbAbgnoMePEYwTwgWcLspDcZMjuzzswGQsh3ePGgiFc/xpt/4yTZgyYWNBRV3XR3hQln2njet5Pb2Nza3snvFvb2Dw6PiscnTS1TRbFBJZeqHRKNnAlsGGY4thOFJA45tsLR3dxvjVFpJsWjmSQYxGQgWMQoMVYKrirdMVGYaMal6BVLXtlbwF0nfkZKkKHeK351+5KmMQpDOdG643uJCaZEGUY5zgrdVGNC6IgMsGOpIDHqYLo4euZeWKXvRlLZEsZdqL8npiTWehKHtjMmZqhXvbn4n9dJTXQbTJlIUoOCLhdFKXeNdOcJuH2mkBo+sYRQxeytLh0SRaixORVsCP7qy+ukWSn71+XqQ7VUq2Zx5OEMzuESfLiBGtxDHRpA4Qme4RXenLHz4rw7H8vWnJPNnMIfOJ8/lsaR9Q==</latexit> 2" <latexit sha1_base64="TV4Oif8hMVSbBDOXM8sKx/5hvIg=">AAAB9HicbVDLSgNBEOyNrxhfUY9eFoPgxbCrQT0GvHiMYB6QLGF20psMmZ1ZZ2YDIeQ7vHhQxKsf482/cZLsQRMLGoqqbrq7woQzbTzv28mtrW9sbuW3Czu7e/sHxcOjhpapolinkkvVColGzgTWDTMcW4lCEoccm+HwbuY3R6g0k+LRjBMMYtIXLGKUGCsFF1edEVGYaMal6BZLXtmbw10lfkZKkKHWLX51epKmMQpDOdG67XuJCSZEGUY5TgudVGNC6JD0sW2pIDHqYDI/euqeWaXnRlLZEsadq78nJiTWehyHtjMmZqCXvZn4n9dOTXQbTJhIUoOCLhZFKXeNdGcJuD2mkBo+toRQxeytLh0QRaixORVsCP7yy6ukcVn2r8uVh0qpWsniyMMJnMI5+HADVbiHGtSBwhM8wyu8OSPnxXl3PhatOSebOYY/cD5/AJhVkfY=</latexit> 3" <latexit sha1_base64="mMcLk3ssuY4tb5iD3uWaAEU6bSQ=">AAACD3icbVDNS8MwHE39nPOr6tFLcFM8jXYM9SIMvHic4D5gLSVNsy0sTUuSCqXsP/Div+LFgyJevXrzvzHtetDNB4HHe79f8vL8mFGpLOvbWFldW9/YrGxVt3d29/bNg8OejBKBSRdHLBIDH0nCKCddRRUjg1gQFPqM9P3pTe73H4iQNOL3Ko2JG6IxpyOKkdKSZ57VM6e4JcMp4jMnRGri+7Dj2TN4DZ2AMIU8q+6ZNathFYDLxC5JDZToeOaXE0Q4CQlXmCEph7YVKzdDQlHMyKzqJJLECE/RmAw15Sgk0s2KJDN4qpUAjiKhD1ewUH9vZCiUMg19PZnnlYteLv7nDRM1unIzyuNEEY7nD40SBlUE83JgQAXBiqWaICyozgrxBAmEla6wqkuwF7+8THrNhn3RaN01a+1WWUcFHIMTcA5scAna4BZ0QBdg8AiewSt4M56MF+Pd+JiPrhjlzhH4A+PzB62Zm7o=</latexit> P 1 = 0 <latexit sha1_base64="3wniU+ebNUb870goZqkhTRFdIAE=">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</latexit> c(x 0) <latexit sha1_base64="ewySohb7cgEbB0upHtwIysKQQps=">AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4Kkkp6rHgxWML9gPaUDbbSbt2swm7G6GE/gIvHhTx6k/y5r9x2+agrQ8GHu/NMDMvSATXxnW/nY3Nre2d3cJecf/g8Oi4dHLa1nGqGLZYLGLVDahGwSW2DDcCu4lCGgUCO8Hkbu53nlBpHssHM03Qj+hI8pAzaqzUdAelsltxFyDrxMtJGXI0BqWv/jBmaYTSMEG17nluYvyMKsOZwFmxn2pMKJvQEfYslTRC7WeLQ2fk0ipDEsbKljRkof6eyGik9TQKbGdEzVivenPxP6+XmvDWz7hMUoOSLReFqSAmJvOvyZArZEZMLaFMcXsrYWOqKDM2m6INwVt9eZ20qxXvulJrVsv1Wh5HAc7hAq7Agxuowz00oAUMEJ7hFd6cR+fFeXc+lq0bTj5zBn/gfP4AdzGMrQ==</latexit> 0 <latexit sha1_base64="k1AtdIimUWBqygfvFGKKOZEHPFk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE8lUSKeix48VjBfkAaymY7aZdudsPupFBCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXpYIb9LxvZ2Nza3tnt7RX3j84PDqunJy2jco0gxZTQuluRA0ILqGFHAV0Uw00iQR0ovH93O9MQBuu5BNOUwgTOpQ85oyilYLehGpIDRdK9itVr+Yt4K4TvyBVUqDZr3z1BoplCUhkghoT+F6KYU41ciZgVu5lBlLKxnQIgaWSJmDCfHHyzL20ysCNlbYl0V2ovydymhgzTSLbmVAcmVVvLv7nBRnGd2HOZZohSLZcFGfCReXO/3cHXANDMbWEMs3trS4bUU0Z2pTKNgR/9eV10r6u+Te1+mO92qgXcZTIObkgV8Qnt6RBHkiTtAgjijyTV/LmoPPivDsfy9YNp5g5I3/gfP4AuIeRgg==</latexit> " <latexit sha1_base64="CT88sq2YM8AU5WkXJiKOQjZuGwA=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lKUY8FLx4r2FZoQ9lsJ+3SzSbubgol9Hd48aCIV3+MN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RQ2Nre2d4q7pb39g8Oj8vFJW8epYthisYjVY0A1Ci6xZbgR+JgopFEgsBOMb+d+Z4JK81g+mGmCfkSHkoecUWMlv0Z6E6ow0VzEsl+uuFV3AbJOvJxUIEezX/7qDWKWRigNE1Trrucmxs+oMpwJnJV6qcaEsjEdYtdSSSPUfrY4ekYurDIgYaxsSUMW6u+JjEZaT6PAdkbUjPSqNxf/87qpCW/8jMskNSjZclGYCmJiMk+ADLhCZsTUEsoUt7cSNqKKMmNzKtkQvNWX10m7VvWuqvX7eqVRz+MowhmcwyV4cA0NuIMmtIDBEzzDK7w5E+fFeXc+lq0FJ585hT9wPn8AgoiR6A==</latexit> 2" <latexit sha1_base64="wi5TWe8g0FbVaEP0bKa26C1fuDw=">AAAB9HicbVBNS8NAEJ3Ur1q/qh69LBbBU0m0qMeCF48VbC20oWy2k3bpZhN3N4US+ju8eFDEqz/Gm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCopeNUMWyyWMSqHVCNgktsGm4EthOFNAoEPgaj25n/OEaleSwfzCRBP6IDyUPOqLGSf0m6Y6ow0VzEsleuuFV3DrJKvJxUIEejV/7q9mOWRigNE1Trjucmxs+oMpwJnJa6qcaEshEdYMdSSSPUfjY/ekrOrNInYaxsSUPm6u+JjEZaT6LAdkbUDPWyNxP/8zqpCW/8jMskNSjZYlGYCmJiMkuA9LlCZsTEEsoUt7cSNqSKMmNzKtkQvOWXV0nroupdVWv3tUq9lsdRhBM4hXPw4BrqcAcNaAKDJ3iGV3hzxs6L8+58LFoLTj5zDH/gfP4AhBiR6Q==</latexit> 3" <latexit sha1_base64="uXKMM5aIxQypveRJvCOybOsJoq0=">AAAB83icbVBNSwMxEM3Wr1q/qh69BIvgxbIrRT0WvHisYD+gu5RsOtuGZpOQZAul9G948aCIV/+MN/+NabsHbX0w8Hhvhpl5seLMWN//9gobm1vbO8Xd0t7+weFR+fikZWSmKTSp5FJ3YmKAMwFNyyyHjtJA0phDOx7dz/32GLRhUjzZiYIoJQPBEkaJdVJ4FY6JBmUYl6JXrvhVfwG8ToKcVFCORq/8FfYlzVIQlnJiTDfwlY2mRFtGOcxKYWZAEToiA+g6KkgKJpoubp7hC6f0cSK1K2HxQv09MSWpMZM0dp0psUOz6s3F/7xuZpO7aMqEyiwIulyUZBxbiecB4D7TQC2fOEKoZu5WTIdEE2pdTCUXQrD68jppXVeDm2rtsVap1/I4iugMnaNLFKBbVEcPqIGaiCKFntErevMy78V79z6WrQUvnzlFf+B9/gAjsZG5</latexit> " <latexit sha1_base64="sbyI3f0EeArzzLyAfPSEEPXZngw=">AAAB9HicbVDLSgNBEOyNrxhfUY9eFoPgxbAbgnoMePEYwTwgWcLspDcZMjuzzswGQsh3ePGgiFc/xpt/4yTZgyYWNBRV3XR3hQln2njet5Pb2Nza3snvFvb2Dw6PiscnTS1TRbFBJZeqHRKNnAlsGGY4thOFJA45tsLR3dxvjVFpJsWjmSQYxGQgWMQoMVYKrirdMVGYaMal6BVLXtlbwF0nfkZKkKHeK351+5KmMQpDOdG643uJCaZEGUY5zgrdVGNC6IgMsGOpIDHqYLo4euZeWKXvRlLZEsZdqL8npiTWehKHtjMmZqhXvbn4n9dJTXQbTJlIUoOCLhdFKXeNdOcJuH2mkBo+sYRQxeytLh0SRaixORVsCP7qy+ukWSn71+XqQ7VUq2Zx5OEMzuESfLiBGtxDHRpA4Qme4RXenLHz4rw7H8vWnJPNnMIfOJ8/lsaR9Q==</latexit> 2" <latexit sha1_base64="TV4Oif8hMVSbBDOXM8sKx/5hvIg=">AAAB9HicbVDLSgNBEOyNrxhfUY9eFoPgxbCrQT0GvHiMYB6QLGF20psMmZ1ZZ2YDIeQ7vHhQxKsf482/cZLsQRMLGoqqbrq7woQzbTzv28mtrW9sbuW3Czu7e/sHxcOjhpapolinkkvVColGzgTWDTMcW4lCEoccm+HwbuY3R6g0k+LRjBMMYtIXLGKUGCsFF1edEVGYaMal6BZLXtmbw10lfkZKkKHWLX51epKmMQpDOdG67XuJCSZEGUY5TgudVGNC6JD0sW2pIDHqYDI/euqeWaXnRlLZEsadq78nJiTWehyHtjMmZqCXvZn4n9dOTXQbTJhIUoOCLhZFKXeNdGcJuD2mkBo+toRQxeytLh0QRaixORVsCP7yy6ukcVn2r8uVh0qpWsniyMMJnMI5+HADVbiHGtSBwhM8wyu8OSPnxXl3PhatOSebOYY/cD5/AJhVkfY=</latexit> 3" <latexit sha1_base64="mMcLk3ssuY4tb5iD3uWaAEU6bSQ=">AAACD3icbVDNS8MwHE39nPOr6tFLcFM8jXYM9SIMvHic4D5gLSVNsy0sTUuSCqXsP/Div+LFgyJevXrzvzHtetDNB4HHe79f8vL8mFGpLOvbWFldW9/YrGxVt3d29/bNg8OejBKBSRdHLBIDH0nCKCddRRUjg1gQFPqM9P3pTe73H4iQNOL3Ko2JG6IxpyOKkdKSZ57VM6e4JcMp4jMnRGri+7Dj2TN4DZ2AMIU8q+6ZNathFYDLxC5JDZToeOaXE0Q4CQlXmCEph7YVKzdDQlHMyKzqJJLECE/RmAw15Sgk0s2KJDN4qpUAjiKhD1ewUH9vZCiUMg19PZnnlYteLv7nDRM1unIzyuNEEY7nD40SBlUE83JgQAXBiqWaICyozgrxBAmEla6wqkuwF7+8THrNhn3RaN01a+1WWUcFHIMTcA5scAna4BZ0QBdg8AiewSt4M56MF+Pd+JiPrhjlzhH4A+PzB62Zm7o=</latexit> P 1 = 0 <latexit sha1_base64="OouAttfCJAoNLTD8gsK8aXHM4E4=">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</latexit> c(x 0) = |x 0| <latexit sha1_base64="CtT1dYNIN0pRGjl7slXvufGidxg=">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</latexit> Wc(Q, 0) = E x⇠Q [c(x 0)] <latexit sha1_base64="j1dkJnJ9HkATyl1idz8INddB+jM=">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</latexit> Bc " ( 0) = {Q : E x⇠Q [c(x 0)]  "}
  5. Capturing of OT Ambiguity Sets from Data <latexit sha1_base64="/+FRpYVTL/Q5DmLt2ZGqfvDazEI=">AAACAnicbVDLSsNAFL3xWesr6krcDBbBVUmkqMuCG5cV7AOaUCbTaTt0MgkzEyGE4sZfceNCEbd+hTv/xkmahbYeGDicc19zgpgzpR3n21pZXVvf2KxsVbd3dvf27YPDjooSSWibRDySvQArypmgbc00p71YUhwGnHaD6U3udx+oVCwS9zqNqR/isWAjRrA20sA+zrxiSEZSLGZeiPUkCFBrIGYDu+bUnQJombglqUGJ1sD+8oYRSUIqNOFYqb7rxNrPsNSMcDqreomiMSZTPKZ9QwUOqfKzYvsMnRlliEaRNE9oVKi/OzIcKpWGganMb1SLXi7+5/UTPbr2MybiRFNB5otGCUc6QnkeaMgkJZqnhmAimbkVkQmWmGiTWtWE4C5+eZl0LuruZb1x16g1G2UcFTiBUzgHF66gCbfQgjYQeIRneIU368l6sd6tj3npilX2HMEfWJ8/8cqXvQ==</latexit> P

    n <latexit sha1_base64="k1AtdIimUWBqygfvFGKKOZEHPFk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE8lUSKeix48VjBfkAaymY7aZdudsPupFBCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXpYIb9LxvZ2Nza3tnt7RX3j84PDqunJy2jco0gxZTQuluRA0ILqGFHAV0Uw00iQR0ovH93O9MQBuu5BNOUwgTOpQ85oyilYLehGpIDRdK9itVr+Yt4K4TvyBVUqDZr3z1BoplCUhkghoT+F6KYU41ciZgVu5lBlLKxnQIgaWSJmDCfHHyzL20ysCNlbYl0V2ovydymhgzTSLbmVAcmVVvLv7nBRnGd2HOZZohSLZcFGfCReXO/3cHXANDMbWEMs3trS4bUU0Z2pTKNgR/9eV10r6u+Te1+mO92qgXcZTIObkgV8Qnt6RBHkiTtAgjijyTV/LmoPPivDsfy9YNp5g5I3/gfP4AuIeRgg==</latexit> " <latexit sha1_base64="gzBBd/K4jjBYVFimyExo36VKlsI=">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</latexit> P n = 1 n n X i=1 zi <latexit sha1_base64="sjlPuQ12qw6GUmCDlxYS0CfJcIU=">AAAB/HicbVDLSsNAFL3xWesr2qWbwSK4KokUdVlw47KCfUAbwmQ6bYdOJmFmIsQQf8WNC0Xc+iHu/BunaRbaeuDC4Zx75849QcyZ0o7zba2tb2xubVd2qrt7+weH9tFxV0WJJLRDIh7JfoAV5UzQjmaa034sKQ4DTnvB7Gbu9x6oVCwS9zqNqRfiiWBjRrA2km/XsmxYvJKRFIv80Xfz3LfrTsMpgFaJW5I6lGj79tdwFJEkpEITjpUauE6svQxLzQineXWYKBpjMsMTOjBU4JAqLyvW5ujMKCM0jqQpoVGh/p7IcKhUGgamM8R6qpa9ufifN0j0+NrLmIgTTQVZLBonHOkIzZNAIyYp0Tw1BBPJzF8RmWKJiTZ5VU0I7vLJq6R70XAvG827Zr3VLOOowAmcwjm4cAUtuIU2dIBACs/wCm/Wk/VivVsfi9Y1q5ypwR9Ynz/PZJV8</latexit> z1 <latexit sha1_base64="r1rPK4D6ooSH9Lkti/dEyALfk9o=">AAAB/HicbVDLSsNAFL2pr1pf0S7dBIvgqiSlqMuCG5cV7APaECbTSTt0MgkzEyGG+CtuXCji1g9x5984TbPQ1gMXDufcO3fu8WNGpbLtb6Oysbm1vVPdre3tHxwemccnfRklApMejlgkhj6ShFFOeooqRoaxICj0GRn485uFP3ggQtKI36s0Jm6IppwGFCOlJc+sZ9m4eCXDKeL5o9fKc89s2E27gLVOnJI0oETXM7/GkwgnIeEKMyTlyLFj5WZIKIoZyWvjRJIY4TmakpGmHIVEulmxNrfOtTKxgkjo4soq1N8TGQqlTENfd4ZIzeSqtxD/80aJCq7djPI4UYTj5aIgYZaKrEUS1oQKghVLNUFYUP1XC8+QQFjpvGo6BGf15HXSbzWdy2b7rt3otMs4qnAKZ3ABDlxBB26hCz3AkMIzvMKb8WS8GO/Gx7K1YpQzdfgD4/MH0OqVfQ==</latexit> z2 <latexit sha1_base64="Xpi8enVimMC1uI6NqNsyZodaZlQ=">AAAB/HicbVDLSsNAFL3xWesr2qWbwSK4KokUdVlw47KCfUAbwmQ6bYdOJmFmIsQQf8WNC0Xc+iHu/BunaRbaeuDC4Zx75849QcyZ0o7zba2tb2xubVd2qrt7+weH9tFxV0WJJLRDIh7JfoAV5UzQjmaa034sKQ4DTnvB7Gbu9x6oVCwS9zqNqRfiiWBjRrA2km/XsmxYvJKRFIv80Rd57tt1p+EUQKvELUkdSrR9+2s4ikgSUqEJx0oNXCfWXoalZoTTvDpMFI0xmeEJHRgqcEiVlxVrc3RmlBEaR9KU0KhQf09kOFQqDQPTGWI9VcveXPzPGyR6fO1lTMSJpoIsFo0TjnSE5kmgEZOUaJ4agolk5q+ITLHERJu8qiYEd/nkVdK9aLiXjeZds95qlnFU4ARO4RxcuIIW3EIbOkAghWd4hTfryXqx3q2PReuaVc7U4A+szx8sYZW5</latexit> zn <latexit sha1_base64="GUD9jfdS/NnObjoPykcSdja/fbk=">AAACIHicbVDNS8MwHE39nPNr6tFLcAjzMloZTgRh6MXjBPcBay1plm1haVqSdFBK/xQv/itePCiiN/1rTLcKuvkg8Hjv/ZJfnhcyKpVpfhpLyyura+uFjeLm1vbObmlvvy2DSGDSwgELRNdDkjDKSUtRxUg3FAT5HiMdb3yd+Z0JEZIG/E7FIXF8NOR0QDFSWnJLddtHauR58MpN7AkSJJSUBTy9x5XEnl6f4Bjx9CfWdHl6Ai8u3VLZrJpTwEVi5aQMcjTd0ofdD3DkE64wQ1L2LDNUToKEopiRtGhHkoQIj9GQ9DTlyCfSSaYbpPBYK304CIQ+XMGp+nsiQb6Use/pZLannPcy8T+vF6nBuZNQHkaKcDx7aBAxqAKYtQX7VBCsWKwJwoLqXSEeIYGw0p0WdQnW/JcXSfu0ap1Va7e1cqOW11EAh+AIVIAF6qABbkATtAAGD+AJvIBX49F4Nt6M91l0ychnDsAfGF/fd8mjtA==</latexit> Bc " (P n) := <latexit sha1_base64="ytn4aQC5e4ThEpUQE7xeyarrrmE=">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</latexit> 2. " = O ⇣ n 1/2 ⌘ <latexit sha1_base64="t3e4yy/vcAiCQEXqJ7GF/FasX4s=">AAACCnicbVC7SgNBFJ2Nrxhfq5Y2o0GwCrshqGXAxjKCeUASwuzs3WTI7OwyDyEsqW38FRsLRWz9Ajv/xkmyhSYeuHA4517uvSdIOVPa876dwtr6xuZWcbu0s7u3f+AeHrVUYiSFJk14IjsBUcCZgKZmmkMnlUDigEM7GN/M/PYDSMUSca8nKfRjMhQsYpRoKw3c056KIhIzPsFaGsCSqTE2aQoSB4kRIYQDt+xVvDnwKvFzUkY5GgP3qxcm1MQgNOVEqa7vpbqfEakZ5TAt9YyClNAxGULXUkFiUP1s/soUn1slxFEibQmN5+rviYzESk3iwHbGRI/UsjcT//O6RkfX/YyJ1GgQdLEoMhzrBM9ywSGTQLWNIWSESmZvxXREJKHapleyIfjLL6+SVrXiX1Zqd9VyvZbHUUQn6AxdIB9doTq6RQ3URBQ9omf0it6cJ+fFeXc+Fq0FJ585Rn/gfP4AZRCaqw==</latexit> true risk upper bounded <latexit sha1_base64="c/eM463Z3P4PakRLS0J6k683uiQ=">AAACBXicbVBNS8NAEN3Ur1q/oh71sFgETyUpRT0WvHisYD+gDWWz2bRLN5uwOxFC6MWLf8WLB0W8+h+8+W/ctjlo64OBx3szzMzzE8E1OM63VVpb39jcKm9Xdnb39g/sw6OOjlNFWZvGIlY9n2gmuGRt4CBYL1GMRL5gXX9yM/O7D0xpHst7yBLmRWQkecgpASMN7dOBDkMScZFhP8MwZljFfqoBK64nQ7vq1Jw58CpxC1JFBVpD+2sQxDSNmAQqiNZ910nAy4kCTgWbVgapZgmhEzJifUMliZj28vkXU3xulACHsTIlAc/V3xM5ibTOIt90RgTGetmbif95/RTCay/nMkmBSbpYFKYCQ4xnkeCAK0bBJBBwQhU3t2I6JopQMMFVTAju8surpFOvuZe1xl292mwUcZTRCTpDF8hFV6iJblELtRFFj+gZvaI368l6sd6tj0VrySpmjtEfWJ8/SoyYZw==</latexit> by the robust risk <latexit sha1_base64="9Gk8xbQ2RXVcd8cfIlNTCo4CoSM=">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</latexit> E z⇠P ? [`(z)]  sup Q2Bc " (P n) E z⇠Q [`(z)] + O(n 1) <latexit sha1_base64="tI1h8NNTfPFX3f/KTHSua0TMKbY=">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</latexit> 1. " = O ⇣ n 1/max{2,d} ⌘ <latexit sha1_base64="/+FRpYVTL/Q5DmLt2ZGqfvDazEI=">AAACAnicbVDLSsNAFL3xWesr6krcDBbBVUmkqMuCG5cV7AOaUCbTaTt0MgkzEyGE4sZfceNCEbd+hTv/xkmahbYeGDicc19zgpgzpR3n21pZXVvf2KxsVbd3dvf27YPDjooSSWibRDySvQArypmgbc00p71YUhwGnHaD6U3udx+oVCwS9zqNqR/isWAjRrA20sA+zrxiSEZSLGZeiPUkCFBrIGYDu+bUnQJombglqUGJ1sD+8oYRSUIqNOFYqb7rxNrPsNSMcDqreomiMSZTPKZ9QwUOqfKzYvsMnRlliEaRNE9oVKi/OzIcKpWGganMb1SLXi7+5/UTPbr2MybiRFNB5otGCUc6QnkeaMgkJZqnhmAimbkVkQmWmGiTWtWE4C5+eZl0LuruZb1x16g1G2UcFTiBUzgHF66gCbfQgjYQeIRneIU368l6sd6tj3npilX2HMEfWJ8/8cqXvQ==</latexit> P n <latexit sha1_base64="k1AtdIimUWBqygfvFGKKOZEHPFk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE8lUSKeix48VjBfkAaymY7aZdudsPupFBCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXpYIb9LxvZ2Nza3tnt7RX3j84PDqunJy2jco0gxZTQuluRA0ILqGFHAV0Uw00iQR0ovH93O9MQBuu5BNOUwgTOpQ85oyilYLehGpIDRdK9itVr+Yt4K4TvyBVUqDZr3z1BoplCUhkghoT+F6KYU41ciZgVu5lBlLKxnQIgaWSJmDCfHHyzL20ysCNlbYl0V2ovydymhgzTSLbmVAcmVVvLv7nBRnGd2HOZZohSLZcFGfCReXO/3cHXANDMbWEMs3trS4bUU0Z2pTKNgR/9eV10r6u+Te1+mO92qgXcZTIObkgV8Qnt6RBHkiTtAgjijyTV/LmoPPivDsfy9YNp5g5I3/gfP4AuIeRgg==</latexit> " <latexit sha1_base64="alKyvyyY1vm4iT2RhDTOYFUgUDw=">AAACB3icbVDLSsNAFL3xWesr6lKQwSK4KokUdVlw47KCfUATymQ6bYdOJmFmIoTQnRt/xY0LRdz6C+78GydpFtp6YOBwzr0zZ04Qc6a043xbK6tr6xubla3q9s7u3r59cNhRUSIJbZOIR7IXYEU5E7Stmea0F0uKw4DTbjC9yf3uA5WKReJepzH1QzwWbMQI1kYa2CeZV1ySkRSLmRdiPQkC1Bp4SmM5QwO75tSdAmiZuCWpQYnWwP7yhhFJQio04VipvuvE2s+w1IxwOqt6iaIxJlM8pn1DBQ6p8rMiwgydGWWIRpE0R2hUqL83MhwqlYaBmcyDqkUvF//z+okeXfsZE3GiqSDzh0YJRzpCeSloyCQlmqeGYCKZyYrIBEtMtKmuakpwF7+8TDoXdfey3rhr1JqNso4KHMMpnIMLV9CEW2hBGwg8wjO8wpv1ZL1Y79bHfHTFKneO4A+szx+VgJm3</latexit> P ? <latexit sha1_base64="SCH5FQ0RUjNJwD71vRokV8OPiwQ=">AAACDHicbVDLSsNAFJ34rPVVdelmsAiuSlKKuiy4cVnBPqANZTK5aYdOJmEeQgn9ADf+ihsXirj1A9z5N07aLLT1wMDhnHu5c06Qcqa06347a+sbm1vbpZ3y7t7+wWHl6LijEiMptGnCE9kLiALOBLQ10xx6qQQSBxy6weQm97sPIBVLxL2epuDHZCRYxCjRVhpWqgMVRSRmfIq1NIBDe1KywOQuZkKxELCdcmvuHHiVeAWpogKtYeVrECbUxCA05USpvuem2s+I1IxymJUHRkFK6ISMoG+pIDEoP5uHmeFzq4Q4SqR9QuO5+nsjI7FS0ziwkzHRY7Xs5eJ/Xt/o6NrPmEiNBkEXhyLDsU5w3oyNLoFqW0TICJXM/hXTMZGEattf2ZbgLUdeJZ16zbusNe7q1WajqKOETtEZukAeukJNdItaqI0oekTP6BW9OU/Oi/PufCxG15xi5wT9gfP5AxJom5U=</latexit> true distribution inside <latexit sha1_base64="pGBZdaFv5LMh2cZCm63yHv49Cug=">AAACB3icbVC7SgNBFJ2Nrxhfq5aCDAbBKuyGoJYBGzsj5AVJCLOTu8mQ2Qczd4UlpLPxV2wsFLH1F+z8GyfJFpp4YOBwzj3cuceLpdDoON9Wbm19Y3Mrv13Y2d3bP7APj5o6ShSHBo9kpNoe0yBFCA0UKKEdK2CBJ6HljW9mfusBlBZRWMc0hl7AhqHwBWdopL592tW+zwIhU4ojoHd1arJimAhMqQbs20Wn5MxBV4mbkSLJUOvbX91BxJMAQuSSad1xnRh7E6ZQcAnTQjfREDM+ZkPoGBqyAHRvMr9jSs+NMqB+pMwLkc7V34kJC7ROA89MBgxHetmbif95nQT9695EhHGCEPLFIj+RFCM6K4UOhAKOpoOBYFwJ81fKR0wxjqa6ginBXT55lTTLJfeyVLkvF6uVrI48OSFn5IK45IpUyS2pkQbh5JE8k1fyZj1ZL9a79bEYzVlZ5pj8gfX5A3PqmQI=</latexit> the OT ambiguity set <latexit sha1_base64="/6ffd+bnzOjan8P41DTh4rKKbB4=">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</latexit> P ? 2 Bc " (P n) Statistical guarantees: 5 <latexit sha1_base64="kPiweoemMIy1foxf2AT3pE7hZjM=">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</latexit> sample distribution P n <latexit sha1_base64="RI3SrSWrgNIM/oUPTZsCun1gEM0=">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</latexit> true distribution P ? <latexit sha1_base64="N/vn0U4LNSeUxJfvcvfYcLYsd7o=">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</latexit> ambiguity set <latexit sha1_base64="GUD9jfdS/NnObjoPykcSdja/fbk=">AAACIHicbVDNS8MwHE39nPNr6tFLcAjzMloZTgRh6MXjBPcBay1plm1haVqSdFBK/xQv/itePCiiN/1rTLcKuvkg8Hjv/ZJfnhcyKpVpfhpLyyura+uFjeLm1vbObmlvvy2DSGDSwgELRNdDkjDKSUtRxUg3FAT5HiMdb3yd+Z0JEZIG/E7FIXF8NOR0QDFSWnJLddtHauR58MpN7AkSJJSUBTy9x5XEnl6f4Bjx9CfWdHl6Ai8u3VLZrJpTwEVi5aQMcjTd0ofdD3DkE64wQ1L2LDNUToKEopiRtGhHkoQIj9GQ9DTlyCfSSaYbpPBYK304CIQ+XMGp+nsiQb6Use/pZLannPcy8T+vF6nBuZNQHkaKcDx7aBAxqAKYtQX7VBCsWKwJwoLqXSEeIYGw0p0WdQnW/JcXSfu0ap1Va7e1cqOW11EAh+AIVIAF6qABbkATtAAGD+AJvIBX49F4Nt6M91l0ychnDsAfGF/fd8mjtA==</latexit> Bc " (P n) :=
  6. Outline Member: °0.50 °0.25 0.00 0.25 0.50 0.75 1.00 1.25

    1.50 x1 °1.0 °0.5 0.0 0.5 1.0 1.5 x2 " = 0.0, cost = 1.13 " = 0.1, cost = 1.28 " = 0.3, cost = 1.39 <latexit sha1_base64="eyzQ2PB/LyqwqRrEmBnoodab6WI=">AAACD3icbVC7SgNBFJ31bXxFLW0Gg2Jj2A2ilqIWdomSqJCEcHcyG4fMzC4zd4Ww5A9s/BUbC0Vsbe38GyePQqMHBg7n3MfcEyZSWPT9L29qemZ2bn5hMbe0vLK6ll/fuLZxahivsVjG5jYEy6XQvIYCJb9NDAcVSn4Tds8G/s09N1bEuoq9hDcVdLSIBAN0Uiu/27BRBErIXlZOUCiQtGpA2yQ2uH/qBrfp+VW538oX/KI/BP1LgjEpkDEqrfxnox2zVHGNTIK19cBPsJmBQcEk7+caqeUJsC50eN1RDYrbZja8p0930sHeKDbuaaRD9WdHBsrangpdpQK8s5PeQPzPq6cYHTczoZMUuWajRVEqKcZ0EA5tC8MZyp4jwIxwf6XsDgwwdBHmXAjB5Ml/yXWpGBwWDy5LhZPSOI4FskW2yR4JyBE5IRekQmqEkQfyRF7Iq/foPXtv3vuodMob92ySX/A+vgEZ8Jyh</latexit> Optimal Transport-Based DRO <latexit sha1_base64="rtt5AGb5mqfSrnGR9IS8KqPkeck=">AAACF3icbVA9T8MwEHX4LOWrwMhi0SIxRUmFgLGIhbFIFJDaqnLcS2vVcYJ9AUVR/wULf4WFAYRYYePf4JYO0PKkk57eu7PvXpBIYdDzvpy5+YXFpeXCSnF1bX1js7S1fWXiVHNo8FjG+iZgBqRQ0ECBEm4SDSwKJFwHg7ORf30H2ohYXWKWQDtiPSVCwRlaqVNyWyYMWSRklt8L7NNWqrqgR6/lcJsyKTAbUhHSymmFCjPslMqe641BZ4k/IWUyQb1T+mx1Y55GoJBLZkzT9xJs50yj4BKGxVZqIGF8wHrQtFSxCEw7H981pPtW6dIw1rYU0rH6eyJnkTFZFNjOiGHfTHsj8T+vmWJ40s6FSlIExX8+ClNJMaajkGhXaOAoM0sY18LuSnmfacbRRlm0IfjTJ8+Sq6rrH7mHF9VyzZ3EUSC7ZI8cEJ8ckxo5J3XSIJw8kCfyQl6dR+fZeXPef1rnnMnMDvkD5+MbkM+gEQ==</latexit> with equality if A is <latexit sha1_base64="upqbFr25ug8f1wDX2YT5wSoHeow=">AAACAnicbVDLSgMxFM34rPU16krcBIvgxmGmiLosuHFZwT6gHUomzbSheQxJRhmG4sZfceNCEbd+hTv/xrSdhbYeCBzOuZebc6KEUW18/9tZWl5ZXVsvbZQ3t7Z3dt29/aaWqcKkgSWTqh0hTRgVpGGoYaSdKIJ4xEgrGl1P/NY9UZpKcWeyhIQcDQSNKUbGSj33sKvjGHHKsjxOGYNKPpwpJEbeuOdWfM+fAi6SoCAVUKDec7+6fYlTToTBDGndCfzEhDlShmJGxuVuqkmC8AgNSMdSgTjRYT6NMIYnVunDWCr7hIFT9fdGjrjWGY/sJEdmqOe9ifif10lNfBXmVCSpIQLPDtmk0Eg46QP2qSLYsMwShBW1f4V4iBTCxrZWtiUE85EXSbPqBRfe+W21UvOKOkrgCByDUxCAS1ADN6AOGgCDR/AMXsGb8+S8OO/Ox2x0ySl2DsAfOJ8/kJqXeQ==</latexit> full row-rank. <latexit sha1_base64="Yeoby0hZxxYD5hokCKO4s415DqU=">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</latexit> A# Bc " (P n) ✓ Bc A† " (A# P n) <latexit sha1_base64="7zV5f8g/OYdJ674qK+e9j2D49ng=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0hE1GPBi8eK9gPaUDbbTbt0swm7E6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z+oHh61TJJpxpsskYnuhNRwKRRvokDJO6nmNA4lb4fj25nffuLaiEQ94iTlQUyHSkSCUbTSQw+TfrXmud4cZJX4BalBgUa/+tUbJCyLuUImqTFd30sxyKlGwSSfVnqZ4SllYzrkXUsVjbkJ8vmpU3JmlQGJEm1LIZmrvydyGhsziUPbGVMcmWVvJv7ndTOMboJcqDRDrthiUZRJggmZ/U0GQnOGcmIJZVrYWwkbUU0Z2nQqNgR/+eVV0rpw/Sv38v6yVneLOMpwAqdwDj5cQx3uoAFNYDCEZ3iFN0c6L86787FoLTnFzDH8gfP5A1vTjcw=</latexit> ! <latexit sha1_base64="YWZkzY+Np26HX231yvTSAwOKTOY=">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</latexit> P n = 1 n n X i=1 zi <latexit sha1_base64="fzNi5+RjOeTdffbis6+8RfeDMgI=">AAAB/HicbVDLSsNAFL2pr1pf0S7dBIvgKiQi6rLgxmUF+4A2hMl00g6dTMLMRIgh/oobF4q49UPc+TdO0yy09cCFwzn3zp17goRRqRzn26itrW9sbtW3Gzu7e/sH5uFRT8apwKSLYxaLQYAkYZSTrqKKkUEiCIoCRvrB7Gbu9x+IkDTm9ypLiBehCachxUhpyTebeT4qX8lxhnjx6LtF4Zstx3ZKWKvErUgLKnR882s0jnEaEa4wQ1IOXSdRXo6EopiRojFKJUkQnqEJGWrKUUSkl5drC+tUK2MrjIUurqxS/T2Ro0jKLAp0Z4TUVC57c/E/b5iq8NrLKU9SRTheLApTZqnYmidhjakgWLFME4QF1X+18BQJhJXOq6FDcJdPXiW9c9u9tC/uLlptu4qjDsdwAmfgwhW04RY60AUMGTzDK7wZT8aL8W58LFprRjXThD8wPn8AzZaVdg==</latexit> z1 <latexit sha1_base64="Bzo1tpzEAKxHrwJVmfgGOB5wSOI=">AAAB/HicbVDLSsNAFL2pr1pf0S7dBIvgKiSlqMuCG5cV7APaECbTSTt0MgkzEyGG+CtuXCji1g9x5984bbPQ1gMXDufcO3fuCRJGpXKcb6Oysbm1vVPdre3tHxwemccnPRmnApMujlksBgGShFFOuooqRgaJICgKGOkHs5u5338gQtKY36ssIV6EJpyGFCOlJd+s5/lo8UqOM8SLR79ZFL7ZcGxnAWuduCVpQImOb36NxjFOI8IVZkjKoeskysuRUBQzUtRGqSQJwjM0IUNNOYqI9PLF2sI618rYCmOhiytrof6eyFEkZRYFujNCaipXvbn4nzdMVXjt5ZQnqSIcLxeFKbNUbM2TsMZUEKxYpgnCguq/WniKBMJK51XTIbirJ6+TXtN2L+3WXavRtss4qnAKZ3ABLlxBG26hA13AkMEzvMKb8WS8GO/Gx7K1YpQzdfgD4/MHzxyVdw==</latexit> z2 <latexit sha1_base64="uIZK1E8QH0yZQJykfM65xHa9Ixo=">AAAB/HicbVDLSsNAFL2pr1pf0S7dBIvgKiQi6rLgxmUF+4A2hMl00g6dTMLMRIgh/oobF4q49UPc+TdO0yy09cCFwzn3zp17goRRqRzn26itrW9sbtW3Gzu7e/sH5uFRT8apwKSLYxaLQYAkYZSTrqKKkUEiCIoCRvrB7Gbu9x+IkDTm9ypLiBehCachxUhpyTebeT4qX8lxhnjx6POi8M2WYzslrFXiVqQFFTq++TUaxziNCFeYISmHrpMoL0dCUcxI0RilkiQIz9CEDDXlKCLSy8u1hXWqlbEVxkIXV1ap/p7IUSRlFgW6M0JqKpe9ufifN0xVeO3llCepIhwvFoUps1RszZOwxlQQrFimCcKC6r9aeIoEwkrn1dAhuMsnr5Leue1e2hd3F622XcVRh2M4gTNw4QracAsd6AKGDJ7hFd6MJ+PFeDc+Fq01o5ppwh8Ynz8qk5Wz</latexit> zn <latexit sha1_base64="49NbEvnTrK6L/vuKSqOaUjn2b6U=">AAACIHicbVDNS8MwHE39nPNr6tFLcAjzMloZTgRh6MXjBPcBay1plm1haVqSdFBK/xQv/itePCiiN/1rTLcKuvkg8Hjv/ZJfnhcyKpVpfhpLyyura+uFjeLm1vbObmlvvy2DSGDSwgELRNdDkjDKSUtRxUg3FAT5HiMdb3yd+Z0JEZIG/E7FIXF8NOR0QDFSWnJLddtHauR58MpN7AkSJJSUBTy9x5XEnl6f4Bjx9CfWdHl6Ai8u3VLZrJpTwEVi5aQMcjTd0ofdD3DkE64wQ1L2LDNUToKEopiRtGhHkoQIj9GQ9DTlyCfSSaYbpPBYK304CIQ+XMGp+nsiQb6Use/pZLannPcy8T+vF6nBuZNQHkaKcDx7aBAxqAKYtQX7VBCsWKwJwoLqXSEeIYGw0p0WdQnW/JcXSfu0ap1Va7e1cqOa11EAh+AIVIAF6qABbkATtAAGD+AJvIBX49F4Nt6M91l0ychnDsAfGF/fdfujrg==</latexit> Bc " (P n) := <latexit sha1_base64="KSkEM5slozmmDteUzhYOdAfO3jw=">AAACAnicbVDNS8MwHE39nPOr6km8BIfgqbQy1OPAi8cJ7gPWUtIs3cLStCSpUErx4r/ixYMiXv0rvPnfmHY96OaDwOO931dekDAqlW1/Gyura+sbm42t5vbO7t6+eXDYl3EqMOnhmMViGCBJGOWkp6hiZJgIgqKAkUEwuyn9wQMRksb8XmUJ8SI04TSkGCkt+eZx7lZDcpwhXrgRUtMggF2fF77Zsi27AlwmTk1aoEbXN7/ccYzTiHCFGZJy5NiJ8nIkFMWMFE03lSRBeIYmZKQpRxGRXl5tL+CZVsYwjIV+XMFK/d2Ro0jKLAp0ZXmjXPRK8T9vlKrw2sspT1JFOJ4vClMGVQzLPOCYCoIVyzRBWFB9K8RTJBBWOrWmDsFZ/PIy6V9YzqXVvmu3OlYdRwOcgFNwDhxwBTrgFnRBD2DwCJ7BK3gznowX4934mJeuGHXPEfgD4/MH7/yXtw==</latexit> P n <latexit sha1_base64="6B9iXydmboZUrALXzhqhACdqzFA=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69BIvgKSQi6rHgxWMF+wFtKJvtpl262Q27k0IJ/RlePCji1V/jzX/jts1BWx8MPN6bYWZelApu0Pe/ndLG5tb2Tnm3srd/cHhUPT5pGZVpyppUCaU7ETFMcMmayFGwTqoZSSLB2tH4fu63J0wbruQTTlMWJmQoecwpQSt1exOiWWq4ULJfrfmev4C7ToKC1KBAo1/96g0UzRImkQpiTDfwUwxzopFTwWaVXmZYSuiYDFnXUkkSZsJ8cfLMvbDKwI2VtiXRXai/J3KSGDNNItuZEByZVW8u/ud1M4zvwpzLNEMm6XJRnAkXlTv/3x1wzSiKqSWEam5vdemIaELRplSxIQSrL6+T1pUX3HjXj9e1ulfEUYYzOIdLCOAW6vAADWgCBQXP8ApvDjovzrvzsWwtOcXMKfyB8/kDtrmRfA==</latexit> " <latexit sha1_base64="XZA6MyOoGJxHcD0Q9JrJduLm4cg=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUhE1GXBjcsK9gFtCJPppB06mYSZiVhDfsWNC0Xc+iPu/BunbRbaeuDC4Zx75849YcqZ0q77bVXW1jc2t6rbtZ3dvf0D+7DeUUkmCW2ThCeyF2JFORO0rZnmtJdKiuOQ0244uZn53QcqFUvEvZ6m1I/xSLCIEayNFNj1fDB/JCdTLAr0FHhFYDdcx50DrRKvJA0o0Qrsr8EwIVlMhSYcK9X33FT7OZaaEU6L2iBTNMVkgke0b6jAMVV+Pl9boFOjDFGUSFNCo7n6eyLHsVLTODSdMdZjtezNxP+8fqajaz9nIs00FWSxKMo40gmaBYGGTFKi+dQQTCQzf0VkjCUm2sRVMyF4yyevks654106F3cXjaZTxlGFYziBM/DgCppwCy1oA4FHeIZXeLMK68V6tz4WrRWrnDmCP7A+fwBXVZSU</latexit> z1 <latexit sha1_base64="Bzo1tpzEAKxHrwJVmfgGOB5wSOI=">AAAB/HicbVDLSsNAFL2pr1pf0S7dBIvgKiSlqMuCG5cV7APaECbTSTt0MgkzEyGG+CtuXCji1g9x5984bbPQ1gMXDufcO3fuCRJGpXKcb6Oysbm1vVPdre3tHxwemccnPRmnApMujlksBgGShFFOuooqRgaJICgKGOkHs5u5338gQtKY36ssIV6EJpyGFCOlJd+s5/lo8UqOM8SLR79ZFL7ZcGxnAWuduCVpQImOb36NxjFOI8IVZkjKoeskysuRUBQzUtRGqSQJwjM0IUNNOYqI9PLF2sI618rYCmOhiytrof6eyFEkZRYFujNCaipXvbn4nzdMVXjt5ZQnqSIcLxeFKbNUbM2TsMZUEKxYpgnCguq/WniKBMJK51XTIbirJ6+TXtN2L+3WXavRtss4qnAKZ3ABLlxBG26hA13AkMEzvMKb8WS8GO/Gx7K1YpQzdfgD4/MHzxyVdw==</latexit> z2 <latexit sha1_base64="uIZK1E8QH0yZQJykfM65xHa9Ixo=">AAAB/HicbVDLSsNAFL2pr1pf0S7dBIvgKiQi6rLgxmUF+4A2hMl00g6dTMLMRIgh/oobF4q49UPc+TdO0yy09cCFwzn3zp17goRRqRzn26itrW9sbtW3Gzu7e/sH5uFRT8apwKSLYxaLQYAkYZSTrqKKkUEiCIoCRvrB7Gbu9x+IkDTm9ypLiBehCachxUhpyTebeT4qX8lxhnjx6POi8M2WYzslrFXiVqQFFTq++TUaxziNCFeYISmHrpMoL0dCUcxI0RilkiQIz9CEDDXlKCLSy8u1hXWqlbEVxkIXV1ap/p7IUSRlFgW6M0JqKpe9ufifN0xVeO3llCepIhwvFoUps1RszZOwxlQQrFimCcKC6r9aeIoEwkrn1dAhuMsnr5Leue1e2hd3F622XcVRh2M4gTNw4QracAsd6AKGDJ7hFd6MJ+PFeDc+Fq01o5ppwh8Ynz8qk5Wz</latexit> zn <latexit sha1_base64="HXb0Go6Ev8C9Owj8AV3kIlRp728=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1gEVyGRoi4LblxWsA9oQplMJu3QyUyYmQglBPwVNy4Ucet3uPNvnKZZaOuBC4dz7p0794Qpo0q77rdVW1vf2Nyqbzd2dvf2D+zDo54SmcSkiwUTchAiRRjlpKupZmSQSoKSkJF+OL2d+/1HIhUV/EHPUhIkaMxpTDHSRhrZJ7lfPpLjGeIF9FkktCpGdtN13BJwlXgVaYIKnZH95UcCZwnhGjOk1NBzUx3kSGqKGSkafqZIivAUjcnQUI4SooK83FzAc6NEMBbSFNewVH9P5ChRapaEpjNBeqKWvbn4nzfMdHwT5JSnmSYcLxbFGYNawHkWMKKSYM1mhiAsqfkrxBMkEdYmsYYJwVs+eZX0Lh3vymndt5ptp4qjDk7BGbgAHrgGbXAHOqALMMjBM3gFb9aT9WK9Wx+L1ppVzRyDP7A+fwAYIZYq</latexit> . . . <latexit sha1_base64="I2D065PvbdZVZVIHiNcaxK5kqhc=">AAACAXicbVDNS8MwHE3n15xfVS+Cl+AQPJVWhnocePG4gfuAtYwkS7ewNC1JKoxSL/4rXjwo4tX/wpv/jenWg24+CDze+33l4YQzpV3326qsrW9sblW3azu7e/sH9uFRV8WpJLRDYh7LPkaKciZoRzPNaT+RFEWY0x6e3hZ+74FKxWJxr2cJDSI0FixkBGkjDe2TzJ8PyTBPaQ79COkJxrCdD+2667hzwFXilaQOSrSG9pc/ikkaUaEJR0oNPDfRQYakZoTTvOaniiaITNGYDgwVKKIqyObLc3hulBEMY2me0HCu/u7IUKTULMKmsrhQLXuF+J83SHV4E2RMJKmmgiwWhSmHOoZFHHDEJCWazwxBRDJzKyQTJBHRJrSaCcFb/vIq6V463pXTaDfqTaeMowpOwRm4AB64Bk1wB1qgAwh4BM/gFbxZT9aL9W59LEorVtlzDP7A+vwBtVuW/g==</latexit> Q <latexit sha1_base64="yUCl9ISYIkEJW7WVMvg6qABpTjU=">AAACH3icbVDLSiNBFK12fMSMOnFczqYwDOim6RZxXApuXGYgUSEJ4XbldlJYj6bq9jBNkz+ZzfyKGxcOw+DOv5lKzMLXgYLDOfdw656sUNJTkjxGKx9W19Y3GpvNj1vbO59au58vvS2dwJ6wyrrrDDwqabBHkhReFw5BZwqvspvzuX/1A52X1nSpKnCoYWJkLgVQkEatk4HPc9BSVfWA8Cdled2donWoZ/ygZwQ6Ammo4h1nC5gsUofxbNRqJ3GyAH9L0iVpsyU6o9bDYGxFqdGQUOB9P00KGtbgSAqFs+ag9FiAuIEJ9gM1oNEP68V9M/41KGOeWxeeIb5Qnydq0N5XOguTGmjqX3tz8T2vX1J+OqylKUpCI54W5aXiZPm8LD6WDgWpKhAQToa/cjEFB4JCpc1QQvr65Lfk8ihOT+Lj70fts3hZR4N9YfvsgKXsGztjF6zDekywX+yW3bM/0e/oLvob/XsaXYmWmT32AtHjf8pbo/c=</latexit> Theorem (Uncertainty Propagation). <latexit sha1_base64="zeZpCaHTuxwitjkPP+1D1w1GztM=">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</latexit> min ✓2⇥ sup Q2Bc " (P n) E Z⇠Q [`(✓, Z)] <latexit sha1_base64="IT4zwnOS/alJbLDBPwgyz3ghYvs=">AAACEHicbVA9T8MwEHX4LOWrwMhi0SKYqqRCwFiJhbFItEU0VXVxHbBwnMi+IEVRfwILf4WFAYRYGdn4N7htBqA86aSn9+58vhckUhh03S9nbn5hcWm5tFJeXVvf2KxsbXdMnGrG2yyWsb4KwHApFG+jQMmvEs0hCiTvBndnY797z7URsbrELOH9CG6UCAUDtNKgcuCbMIRIyCxvK8Y1glCY0Zo/eTpnGagRva6NBpWqW3cnoLPEK0iVFGgNKp/+MGZpxBUyCcb0PDfBfg4aBZN8VPZTwxNgd3DDe5YqiLjp55OtI7pvlSENY21LIZ2oPydyiIzJosB2RoC35q83Fv/zeimGp/1cqCRFrth0UZhKijEdp0OHQnOGMrMEmBb2r5TdggaGNsOyDcH7e/Is6TTq3nH96KJRbdaLOEpkl+yRQ+KRE9Ik56RF2oSRB/JEXsir8+g8O2/O+7R1zilmdsgvOB/fdr+daQ==</latexit> Uncertainty Z <latexit sha1_base64="Ffs9jUqsfvhwZt1ii+dVmv/sVRg=">AAACE3icbVDLSgMxFM3UV62vqks3wSKIizJTRF0W6sJlFfuAdiiZNNOGZpIhuSOUof/gxl9x40IRt27c+Tem7Sy09UDgcM693JwTxIIbcN1vJ7eyura+kd8sbG3v7O4V9w+aRiWasgZVQul2QAwTXLIGcBCsHWtGokCwVjCqTf3WA9OGK3kP45j5ERlIHnJKwEq94lnXhCGJuBin1/aa5kEyNYgQY3yngsQArikJWolJr1hyy+4MeJl4GSmhDPVe8avbVzSJmAQqiDEdz43BT4kGTgWbFLqJYTGhIzJgHUsliZjx01mmCT6xSh+HStsnAc/U3xspiYwZR4GdjAgMzaI3Ff/zOgmEV37KZZwAk3R+KEwEBoWnBeE+14yCjd/nhGpu/4rpkGhCwdZYsCV4i5GXSbNS9i7K57eVUrWS1ZFHR+gYnSIPXaIqukF11EAUPaJn9IrenCfnxXl3PuajOSfbOUR/4Hz+AL3zn0c=</latexit> Distributionally Robust Control <latexit sha1_base64="zTbFC1XD/UxFkaN7ov05PVfH9Ns=">AAACEnicbVDLSgMxFM34rPU16tJNsBXaTZkpooIIBTcuK9oHdNqSSTNtaJIZkoxShvkGN/6KGxeKuHXlzr8xfSy09cCFwzn3cu89fsSo0o7zbS0tr6yurWc2sptb2zu79t5+XYWxxKSGQxbKpo8UYVSQmqaakWYkCeI+Iw1/eDX2G/dEKhqKOz2KSJujvqABxUgbqWsXPRUEiFM2greIR4wo6F3kveShkxRoMfXSbkIv3bQj8l0755ScCeAicWckB2aodu0vrxfimBOhMUNKtVwn0u0ESU0xI2nWixWJEB6iPmkZKhAnqp1MXkrhsVF6MAilKaHhRP09kSCu1Ij7ppMjPVDz3lj8z2vFOjhvJ1REsSYCTxcFMYM6hON8YI9KgrWJo0cRltTcCvEASYS1STFrQnDnX14k9XLJPS2d3JRzldIsjgw4BEegAFxwBirgGlRBDWDwCJ7BK3iznqwX6936mLYuWbOZA/AH1ucPKLudFA==</latexit> Samples {w(i)}n i=1 <latexit sha1_base64="S+fpjNVn6IFSkG6H8GEGtMyYGwA=">AAACC3icbVDLSgMxFM3UV62vqks3oa0gFMpMERVEqLpxWcE+oB2GTJq2oZkHyR1tGbp346+4caGIW3/AnX9jOu1CWw9c7uGce0nucUPBFZjmt5FaWl5ZXUuvZzY2t7Z3srt7dRVEkrIaDUQgmy5RTHCf1YCDYM1QMuK5gjXcwfXEb9wzqXjg38EoZLZHej7vckpAS042Vxg6MRStMb7Al3joAC7iKxwlvX2OHxwoONm8WTIT4EVizUgezVB1sl/tTkAjj/lABVGqZZkh2DGRwKlg40w7UiwkdEB6rKWpTzym7Di5ZYwPtdLB3UDq8gEn6u+NmHhKjTxXT3oE+mrem4j/ea0Iumd2zP0wAubT6UPdSGAI8CQY3OGSURAjTQiVXP8V0z6RhIKOL6NDsOZPXiT1csk6KR3flvOV0iyONDpAOXSELHSKKugGVVENUfSIntErejOejBfj3fiYjqaM2c4++gPj8weOo5eD</latexit> xt+1 = Axt + But + wt <latexit sha1_base64="62Z7LoFbR9F4wmeNPJ+jA7EO7Vc=">AAAB+nicbVDLSsNAFJ34rPWV6tLNYCu4MSRF1GXBjcuK9gFtCJPppB06eTBzo5TYT3HjQhG3fok7/8ZJm4W2Hhg4nHMv98zxE8EV2Pa3sbK6tr6xWdoqb+/s7u2blYO2ilNJWYvGIpZdnygmeMRawEGwbiIZCX3BOv74Ovc7D0wqHkf3MEmYG5JhxANOCWjJMyu1fkhg5Pv4zsvgzJnWPLNqW/YMeJk4BamiAk3P/OoPYpqGLAIqiFI9x07AzYgETgWblvupYgmhYzJkPU0jEjLlZrPoU3yilQEOYqlfBHim/t7ISKjUJPT1ZJ5TLXq5+J/XSyG4cjMeJSmwiM4PBanAEOO8BzzgklEQE00IlVxnxXREJKGg2yrrEpzFLy+Tdt1yLqzz23q1YRV1lNAROkanyEGXqIFuUBO1EEWP6Bm9ojfjyXgx3o2P+eiKUewcoj8wPn8AvvSS9g==</latexit> S t 1 <latexit sha1_base64="8Yk6fk/TU9Bw9ivzkA2GX4a4g5U=">AAAB+HicbVBNS8NAFNzUr1o/GvXoZbEVPJWkiHosePFY0dZCG8Jmu2mXbjZh90Woob/EiwdFvPpTvPlv3LQ5aOvAwjDzHm92gkRwDY7zbZXW1jc2t8rblZ3dvf2qfXDY1XGqKOvQWMSqFxDNBJesAxwE6yWKkSgQ7CGYXOf+wyNTmsfyHqYJ8yIykjzklICRfLtaH0QExkGA7/wMZnXfrjkNZw68StyC1FCBtm9/DYYxTSMmgQqidd91EvAyooBTwWaVQapZQuiEjFjfUEkipr1sHnyGT40yxGGszJOA5+rvjYxEWk+jwEzmKfWyl4v/ef0Uwisv4zJJgUm6OBSmAkOM8xbwkCtGQUwNIVRxkxXTMVGEgumqYkpwl7+8SrrNhnvROL9t1lqNoo4yOkYn6Ay56BK10A1qow6iKEXP6BW9WU/Wi/VufSxGS1axc4T+wPr8AdvmkoQ=</latexit> S t <latexit sha1_base64="5dFsExeU9LmjTQ+O2BpFHfnJVpY=">AAAB+nicbVDLSsNAFJ34rPWV6tLNYCsIQkiKqMuCG5cV7QPaECbTSTt08mDmRimxn+LGhSJu/RJ3/o2TNgttPTBwOOde7pnjJ4IrsO1vY2V1bX1js7RV3t7Z3ds3KwdtFaeSshaNRSy7PlFM8Ii1gINg3UQyEvqCdfzxde53HphUPI7uYZIwNyTDiAecEtCSZ1Zq/ZDAyPfxnZfBmTOteWbVtuwZ8DJxClJFBZqe+dUfxDQNWQRUEKV6jp2AmxEJnAo2LfdTxRJCx2TIeppGJGTKzWbRp/hEKwMcxFK/CPBM/b2RkVCpSejryTynWvRy8T+vl0Jw5WY8SlJgEZ0fClKBIcZ5D3jAJaMgJpoQKrnOiumISEJBt1XWJTiLX14m7brlXFjnt/VqwyrqKKEjdIxOkYMuUQPdoCZqIYoe0TN6RW/Gk/FivBsf89EVo9g5RH9gfP4Au+aS9A==</latexit> S t+1 <latexit sha1_base64="2hlKh5iX47mQJnId3ZO9wSlDEpU=">AAACCHicbVC7SgNBFJ31GeNr1dLCwSBYhd0gaiMEbCwjmgckS5idvZsMmX0wc1cIIaWNv2JjoYitn2Dn3zhJttDEAwOHc+7hzj1+KoVGx/m2lpZXVtfWCxvFza3tnV17b7+hk0xxqPNEJqrlMw1SxFBHgRJaqQIW+RKa/uB64jcfQGmRxPc4TMGLWC8WoeAMjdS1jzo6DFkk5JDeIUOgV9SERS8TOKQasGuXnLIzBV0kbk5KJEeta391goRnEcTIJdO67TopeiOmUHAJ42In05AyPmA9aBsaswi0N5oeMqYnRglomCjzYqRT9XdixCKth5FvJiOGfT3vTcT/vHaG4aU3EnGaIcR8tijMJMWETlqhgVDA0ZQQCMaVMH+lvM8U42i6K5oS3PmTF0mjUnbPy2e3lVK1nNdRIIfkmJwSl1yQKrkhNVInnDySZ/JK3qwn68V6tz5mo0tWnjkgf2B9/gAq8Zlg</latexit> State = ambiguity set °0.50 °0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 x1 °1.0 °0.5 0.0 0.5 1.0 1.5 x2 " = 0.0, cost = 1.13 " = 0.1, cost = 1.28 " = 0.3, cost = 1.39 ε = 0 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% ***** ***** ***** ***** ***** ***** ***** ***** ***** ***** ε = 0.01 100% 1% 1% 1% 1% 1% 1% 1% 1% 1% ***** ***** ***** ***** ***** ***** ***** ***** ***** ***** ε = 0.1 100% 5% 2% 4% 6% 4% 1% 7% 2% 5% ***** ***** ***** ***** ***** ***** ***** ***** ***** ***** ε = 1 100% 49% 17% 32% 56% 36% 3% 66% 13% 49% ***** ***** ***** ***** ***** ***** ***** ***** ***** ***** ε = 10 <latexit sha1_base64="3ZwfhhT3zSgT9K86tsyQhhXwlGA=">AAACF3icbVBNS8NAEN34WetX1aOXxSLUS0hE1GNBBE9SoVWhKWWznbSLu0ncnYgl9F948a948aCIV735b9zWHPx6MPB4b4aZeWEqhUHP+3Cmpmdm5+ZLC+XFpeWV1cra+rlJMs2hxROZ6MuQGZAihhYKlHCZamAqlHARXh2N/Ysb0EYkcROHKXQU68ciEpyhlboVNzBRxJSQQxog3GIY5c0BJBrUiNZOmRnQ4+tMSBFqkakdt1upeq43Af1L/IJUSYFGt/Ie9BKeKYiRS2ZM2/dS7ORMo+ASRuUgM5AyfsX60LY0ZgpMJ5/8NaLbVunRKNG2YqQT9ftEzpQxQxXaTsVwYH57Y/E/r51hdNjJRZxmCDH/WhRlkmJCxyHRntDA0WbSE4xrYW+lfMA042ijLNsQ/N8v/yXnu66/7+6d7VbrbhFHiWySLVIjPjkgdXJCGqRFOLkjD+SJPDv3zqPz4rx+tU45xcwG+QHn7RMQyp/H</latexit> Theorem (Nash Equilibrium). <latexit sha1_base64="5njVQ1Dr67ba/oILxm9l9Jqv8EM=">AAACDnicbVDLSgMxFM3UV62vUZdugqXgqswUUZcFXbizSl/QlpJJ77ShyWRMMkIp/QI3/oobF4q4de3OvzFtZ6GtBwKHc+7h5p4g5kwbz/t2Miura+sb2c3c1vbO7p67f1DXMlEUalRyqZoB0cBZBDXDDIdmrICIgEMjGF5O/cYDKM1kVDWjGDqC9CMWMkqMlbpuoa3DkAjGR7g6AHx1d4NjJW1aYKYx3CeEYyO7bt4rejPgZeKnJI9SVLruV7snaSIgMpQTrVu+F5vOmCjDKIdJrp1oiAkdkj60LI2IAN0Zz86Z4IJVejiUyr7I4Jn6OzEmQuuRCOykIGagF72p+J/XSkx40RmzKE4MRHS+KEymB+JpN7jHFFBjq+gxQhWzf8V0QBShxjaYsyX4iycvk3qp6J8VT29L+XIxrSOLjtAxOkE+OkdldI0qqIYoekTP6BW9OU/Oi/PufMxHM06aOUR/4Hz+AOoKm1U=</latexit> The DRO problem is equal to <latexit sha1_base64="TnthMt3fANIrt1dolII9mFGLnMI=">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</latexit> max Q2Bc " (P n) min ✓2⇥ E Z⇠Q [`(✓, Z)] <latexit sha1_base64="XliWBmEV0gkxiBicwLTnO6z97rU=">AAACGnicbVDLSgMxFM34rPVVdekmWARXw4yIiitBBF0oirYKbSl3MhkNTTJDckcsQ7/Djb/ixoUi7sSNf2Nau/B1IHA4515O7okyKSwGwYc3Mjo2PjFZmipPz8zOzVcWFus2zQ3jNZbK1FxGYLkUmtdQoOSXmeGgIskvos5e37+44caKVJ9jN+MtBVdaJIIBOqldCZs2SUAJ2aVN5LcYJcX+LahM8t4OPasf0VTTo+PDs3MaA7oc9NuVauAHA9C/JBySKhnipF15a8YpyxXXyCRY2wiDDFsFGBTMxZSbueUZsA5c8YajGhS3rWJwWo+uOiWmSWrc00gH6veNApS1XRW5SQV4bX97ffE/r5Fjst0qhM5y5Jp9BSW5pJjSfk80FoYzdLXEApgR7q+UXYMBhq7Nsish/H3yX1Jf98NNf+N0vbrrD+sokWWyQtZISLbILjkgJ6RGGLkjD+SJPHv33qP34r1+jY54w50l8gPe+ycoVqA4</latexit> Example: SVM on MNIST dataset. <latexit sha1_base64="PJKqAwmKW3xB3z2OapAsscfTMoQ=">AAAB9XicbVDLTsMwEHTKq5RXgSMXiwqJU5QgBBwrceFYJPqQ2lA5zqa16jiR7YBK2v/gwgGEuPIv3Pgb3DQHaBlppdHMrtc7fsKZ0o7zbZVWVtfWN8qbla3tnd296v5BS8WppNCkMY9lxycKOBPQ1Exz6CQSSORzaPuj65nffgCpWCzu9DgBLyIDwUJGiTbSfS9/IZMQTCdPk3615thODrxM3ILUUIFGv/rVC2KaRiA05USprusk2suI1IxymFZ6qYKE0BEZQNdQQSJQXpbvnOITowQ4jKUpoXGu/p7ISKTUOPJNZ0T0UC16M/E/r5vq8MrLmEhSDYLOF4UpxzrGswhwwCRQzceGECqZ+SumQyIJ1SaoignBXTx5mbTObPfCPr89r9XtIo4yOkLH6BS56BLV0Q1qoCaiSKJn9IrerEfrxXq3PuatJauYOUR/YH3+AHzXkxs=</latexit> |z| <latexit sha1_base64="6B9iXydmboZUrALXzhqhACdqzFA=">AAAB8nicbVBNS8NAEJ3Ur1q/qh69BIvgKSQi6rHgxWMF+wFtKJvtpl262Q27k0IJ/RlePCji1V/jzX/jts1BWx8MPN6bYWZelApu0Pe/ndLG5tb2Tnm3srd/cHhUPT5pGZVpyppUCaU7ETFMcMmayFGwTqoZSSLB2tH4fu63J0wbruQTTlMWJmQoecwpQSt1exOiWWq4ULJfrfmev4C7ToKC1KBAo1/96g0UzRImkQpiTDfwUwxzopFTwWaVXmZYSuiYDFnXUkkSZsJ8cfLMvbDKwI2VtiXRXai/J3KSGDNNItuZEByZVW8u/ud1M4zvwpzLNEMm6XJRnAkXlTv/3x1wzSiKqSWEam5vdemIaELRplSxIQSrL6+T1pUX3HjXj9e1ulfEUYYzOIdLCOAW6vAADWgCBQXP8ApvDjovzrvzsWwtOcXMKfyB8/kDtrmRfA==</latexit> " <latexit sha1_base64="ZxV2tVRoUuVcLZBYiTCqqvAyJ4g=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBbBiyGRoh4LXjxWsB/QhLLZTtqlm82yuymU0L/hxYMiXv0z3vw3btsctPXBwOO9GWbmRZIzbTzv2yltbG5t75R3K3v7B4dH1eOTtk4zRaFFU56qbkQ0cCagZZjh0JUKSBJx6ETj+7nfmYDSLBVPZiohTMhQsJhRYqwUXAUTokBqxlPRr9Y811sArxO/IDVUoNmvfgWDlGYJCEM50brne9KEOVGGUQ6zSpBpkISOyRB6lgqSgA7zxc0zfGGVAY5TZUsYvFB/T+Qk0XqaRLYzIWakV725+J/Xy0x8F+ZMyMyAoMtFccaxSfE8ADxgCqjhU0sIVczeiumIKEKNjaliQ/BXX14n7WvXv3Hrj/Vawy3iKKMzdI4ukY9uUQM9oCZqIYokekav6M3JnBfn3flYtpacYuYU/YHz+QMh45Gz</latexit> " <latexit sha1_base64="xOTKx/QrXGOf15d62OjZAHdrX7s=">AAAB83icbVBNS8NAEN3Ur1q/qh69BIvgKSRa1GPBi8cK9gOaUDbbSbt0s7vsbgol9G948aCIV/+MN/+N2zYHbX0w8Hhvhpl5sWRUG9//dkobm1vbO+Xdyt7+weFR9fikrUWmCLSIYEJ1Y6yBUQ4tQw2DrlSA05hBJx7fz/3OBJSmgj+ZqYQoxUNOE0qwsVJ4HU6wAqkpE7xfrfmev4C7ToKC1FCBZr/6FQ4EyVLghjCsdS/wpYlyrAwlDGaVMNMgMRnjIfQs5TgFHeWLm2fuhVUGbiKULW7chfp7Isep1tM0tp0pNiO96s3F/7xeZpK7KKdcZgY4WS5KMuYa4c4DcAdUATFsagkmitpbXTLCChNjY6rYEILVl9dJ+8oLbrz6Y73W8Io4yugMnaNLFKBb1EAPqIlaiCCJntErenMy58V5dz6WrSWnmDlFf+B8/gArPZG5</latexit> 3" <latexit sha1_base64="1U1i8pelsQH9VoSYr4vj5UFC/aQ=">AAAB9HicbVBNSwMxEJ2tX7V+VT16CRbBi8uuFvVY8OKxgv2AdinZNNuGZpM1yRbK0t/hxYMiXv0x3vw3pu0etPXBwOO9GWbmhQln2njet1NYW9/Y3Cpul3Z29/YPyodHTS1TRWiDSC5VO8SaciZowzDDaTtRFMchp61wdDfzW2OqNJPi0UwSGsR4IFjECDZWCi6uumOsaKIZl6JXrniuNwdaJX5OKpCj3it/dfuSpDEVhnCsdcf3EhNkWBlGOJ2WuqmmCSYjPKAdSwWOqQ6y+dFTdGaVPoqksiUMmqu/JzIcaz2JQ9sZYzPUy95M/M/rpCa6DTImktRQQRaLopQjI9EsAdRnihLDJ5Zgopi9FZEhVpgYm1PJhuAvv7xKmpeuf+1WH6qVmpvHUYQTOIVz8OEGanAPdWgAgSd4hld4c8bOi/PufCxaC04+cwx/4Hz+AJaHkfA=</latexit> 3" <latexit sha1_base64="I5ZMuLpnURx4BUTx1fNMhGQea40=">AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0ikqMeCF48t2A9oQ9lsJ+3azSbsboRS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8MBVcG8/7dgobm1vbO8Xd0t7+weFR+fikpZNMMWyyRCSqE1KNgktsGm4EdlKFNA4FtsPx3dxvP6HSPJEPZpJiENOh5BFn1Fip4fXLFc/1FiDrxM9JBXLU++Wv3iBhWYzSMEG17vpeaoIpVYYzgbNSL9OYUjamQ+xaKmmMOpguDp2RC6sMSJQoW9KQhfp7YkpjrSdxaDtjakZ61ZuL/3ndzES3wZTLNDMo2XJRlAliEjL/mgy4QmbExBLKFLe3EjaiijJjsynZEPzVl9dJ68r1r91qo1qpuXkcRTiDc7gEH26gBvdQhyYwQHiGV3hzHp0X5935WLYWnHzmFP7A+fwBdgOMqQ==</latexit> 0 <latexit sha1_base64="aiNFXklOGCKVgXYEC5l+/3m/U/w=">AAACDnicbVDNS8MwHE3n15xfVY9egmPgqbQy1Isw8OJxgvuAtZQ0y7awNC1JKpTSv8CL/4oXD4p49ezN/8a060E3HwQe7/1+ycsLYkalsu1vo7a2vrG5Vd9u7Ozu7R+Yh0d9GSUCkx6OWCSGAZKEUU56iipGhrEgKAwYGQTzm8IfPBAhacTvVRoTL0RTTicUI6Ul32xlbnlJhlPEc+iGSM2CAHZ9J4fX0B0TppBv+2bTtuwScJU4FWmCCl3f/HLHEU5CwhVmSMqRY8fKy5BQFDOSN9xEkhjhOZqSkaYchUR6WZkkhy2tjOEkEvpwBUv190aGQinTMNCTRVy57BXif94oUZMrL6M8ThThePHQJGFQRbDoBo6pIFixVBOEBdVZIZ4hgbDSDTZ0Cc7yl1dJ/9xyLqz2XbvZsao66uAEnIIz4IBL0AG3oAt6AINH8AxewZvxZLwY78bHYrRmVDvH4A+Mzx9Ac5uE</latexit> P 1 = 0 <latexit sha1_base64="tQ7ABm7Gs++U+2vAfL1v0AfaDMk=">AAAB9HicbVBNSwMxEJ31s9avqkcvwSp4KrtF1GNBBI9V3LbQLiWbzbahSXZNsoVS+ju8eFDEqz/Gm//GtN2Dtj4YeLw3w8y8MOVMG9f9dlZW19Y3Ngtbxe2d3b390sFhQyeZItQnCU9UK8Saciapb5jhtJUqikXIaTMc3Ez95pAqzRL5aEYpDQTuSRYzgo2Vgo6OYywYH6Fb/6FbKrsVdwa0TLyclCFHvVv66kQJyQSVhnCsddtzUxOMsTKMcDopdjJNU0wGuEfblkosqA7Gs6Mn6MwqEYoTZUsaNFN/T4yx0HokQtspsOnrRW8q/ue1MxNfB2Mm08xQSeaL4owjk6BpAihiihJjX44YJorZWxHpY4WJsTkVbQje4svLpFGteJeVi/tquXaax1GAYziBc/DgCmpwB3XwgcATPMMrvDlD58V5dz7mrStOPnMEf+B8/gA7c5Go</latexit> EUR <latexit sha1_base64="LpICOOWmSpbTPVam1NaGvdUH5gM=">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</latexit> min PT t=1 `(xt, ut) s.t. supQ2S t CVaRxt ⇠Q (xt 2 X)  0 <latexit sha1_base64="qKDMf2vBKQiNlmy7q9pLPaWaR3c=">AAAB+3icbVDLSsNAFJ3UV62vWJduBlvBVUiKqMtCNy4r2FZoQ5lMJu3QeYSZiVhCf8WNC0Xc+iPu/BunbRbaeuDC4Zx7ufeeKGVUG9//dkobm1vbO+Xdyt7+weGRe1ztapkpTDpYMqkeIqQJo4J0DDWMPKSKIB4x0osmrbnfeyRKUynuzTQlIUcjQROKkbHS0K0OdJIgTtkUtqQ2sI7rQ7fme/4CcJ0EBamBAu2h+zWIJc44EQYzpHU/8FMT5kgZihmZVQaZJinCEzQifUsF4kSH+eL2GTy3SgwTqWwJAxfq74kcca2nPLKdHJmxXvXm4n9ePzPJTZhTkWaGCLxclGQMGgnnQcCYKoKN/TumCCtqb4V4jBTCxsZVsSEEqy+vk27DC668y7tGrekVcZTBKTgDFyAA16AJbkEbdAAGT+AZvII3Z+a8OO/Ox7K15BQzJ+APnM8fxXGTjQ==</latexit> Cost c <latexit sha1_base64="Ee38jRD+dSh/6KJKP93flHAHj+k=">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</latexit> in Bc " (P n) <latexit sha1_base64="jmHmwyLxBYrFTM+DddzQBrFZ/yI=">AAAB+3icbVBNS8NAEJ3Ur1q/Yj16WSyCp5IUUY8FLx4r2A9oQ9lsNu3S3U3Y3Ygl9K948aCIV/+IN/+N2zYHbX0w8Hhvhpl5YcqZNp737ZQ2Nre2d8q7lb39g8Mj97ja0UmmCG2ThCeqF2JNOZO0bZjhtJcqikXIaTec3M797iNVmiXywUxTGgg8kixmBBsrDd3qQMcxFoxPEcGpyRTVQ7fm1b0F0DrxC1KDAq2h+zWIEpIJKg3hWOu+76UmyLEyjHA6qwwyTVNMJnhE+5ZKLKgO8sXtM3RulQjFibIlDVqovydyLLSeitB2CmzGetWbi/95/czEN0HOZJoZKslyUZxxZBI0DwJFTFFi7N8Rw0QxeysiY6wwMTauig3BX315nXQadf+qfnnfqDXrRRxlOIUzuAAfrqEJd9CCNhB4gmd4hTdn5rw4787HsrXkFDMn8AfO5w9QNZSQ</latexit> captures <latexit sha1_base64="+9e9y9hq+5/gMRKRs3XSTKR5sOg=">AAACAXicbVDLSgMxFM3UV62vUTeCm2ARXJWZIuqy4MZlBfuAzlAymUwbmswMyR2xDHXjr7hxoYhb/8Kdf2PazkJbDwQO59yb5JwgFVyD43xbpZXVtfWN8mZla3tnd8/eP2jrJFOUtWgiEtUNiGaCx6wFHATrpooRGQjWCUbXU79zz5TmSXwH45T5kgxiHnFKwEh9+8jTUUQkF2PsAXuAIMoDQeho0rerTs2ZAS8TtyBVVKDZt7+8MKGZZDFQQbTuuU4Kfk4UcCrYpOJlmqXmZjJgPUNjIpn281mCCT41SoijRJkTA56pvzdyIrUey8BMSgJDvehNxf+8XgbRlZ/zOM2AxXT+UJQJDAme1oFDrhgFkz7khCpu/orpkChCwZRWMSW4i5GXSbtecy9q57f1aqNW1FFGx+gEnSEXXaIGukFN1EIUPaJn9IrerCfrxXq3PuajJavYOUR/YH3+AB0gl0M=</latexit> black <latexit sha1_base64="q/yKz8g7EHu8AIXfW3zYl6HZh2A=">AAACAHicbVDLSgMxFM3UV62vqgsXboJFcDXMFFGXBTcuK9gHtKVkMpk2NMkMyR21DLPxV9y4UMStn+HOvzF9LLT1QOBwzr3cnBMkghvwvG+nsLK6tr5R3Cxtbe/s7pX3D5omTjVlDRqLWLcDYpjgijWAg2DtRDMiA8Faweh64rfumTY8VncwTlhPkoHiEacErNQvH3VNFBHJxRh3gT1CEGXmgai8X654rjcFXib+nFTQHPV++asbxjSVTAEVxJiO7yXQy4gGTgXLS93UsITQERmwjqWKSGZ62TRAjk+tEuIo1vYpwFP190ZGpDFjGdhJSWBoFr2J+J/XSSG66mVcJSkwRWeHolRgiPGkDRxyzSjY8CEnVHP7V0yHRBMKtrOSLcFfjLxMmlXXv3DPb6uVmjuvo4iO0Qk6Qz66RDV0g+qogSjK0TN6RW/Ok/PivDsfs9GCM985RH/gfP4AiAqW9Q==</latexit> swan <latexit sha1_base64="seT6Cke5pc5G/44L39YZtxMzj10=">AAACAnicbVC7SgNBFJ31GeNr1UpsBoNgFXaDqGXAxjKCeUA2hNnJ3WTI7IOZu8GwBBt/xcZCEVu/ws6/cZJsoYkHBg7n3MOde/xECo2O822trK6tb2wWtorbO7t7+/bBYUPHqeJQ57GMVctnGqSIoI4CJbQSBSz0JTT94c3Ub45AaRFH9zhOoBOyfiQCwRkaqWsfezoIWCjkmHoID+gHGYwgQj3p2iWn7MxAl4mbkxLJUevaX14v5mlo0lwyrduuk2AnYwoFlzApeqmGhPEh60Pb0IiFoDvZ7IQJPTNKjwaxMi9COlN/JzIWaj0OfTMZMhzoRW8q/ue1UwyuO5mIkhQh4vNFQSopxnTaB+0JBRzN+T3BuBLmr5QPmGIcTWtFU4K7ePIyaVTK7mX54q5SqpbzOgrkhJySc+KSK1Ilt6RG6oSTR/JMXsmb9WS9WO/Wx3x0xcozR+QPrM8fMvKX5Q==</latexit> events <latexit sha1_base64="4dwRHQFr4VsGtDdl3Iw9Z8FG4eA=">AAACAXicbVDLSsNAFJ34rPUVdSO4GSyCq5IUUZcFEdwIFewD2lAmk5t26GQSZiZCCHXjr7hxoYhb/8Kdf+O0zUJbDwwczrmXO+f4CWdKO863tbS8srq2Xtoob25t7+zae/stFaeSQpPGPJYdnyjgTEBTM82hk0ggkc+h7Y+uJn77AaRisbjXWQJeRAaChYwSbaS+fdhTYUgixjN8LUAOMnxL5Ai06tsVp+pMgReJW5AKKtDo21+9IKZpBEJTTpTquk6ivZxIzSiHcbmXKkgIHZEBdA0VJALl5dMEY3xilACHsTRPaDxVf2/kJFIqi3wzGRE9VPPeRPzP66Y6vPRyJpJUg6CzQ2HKsY7xpA4cMAlUm/QBI1Qy81dMh0QSqk1pZVOCOx95kbRqVfe8enZXq9RrRR0ldISO0Sly0QWqoxvUQE1E0SN6Rq/ozXqyXqx362M2umQVOwfoD6zPH1yllso=</latexit> Energy Markets <latexit sha1_base64="MOX5OQPTLeaPNVCeF4uid4mek38=">AAACA3icbVDLSgMxFM34rPU16k43wSK4KjNF1GXBjQuFCvYB7VDupJk2NMkMSUYoQ8GNv+LGhSJu/Ql3/o1pOwttPRA4nHNvknPChDNtPO/bWVpeWV1bL2wUN7e2d3bdvf2GjlNFaJ3EPFatEDTlTNK6YYbTVqIoiJDTZji8mvjNB6o0i+W9GSU0ENCXLGIEjJW67mFHRxEIxkf4FsjA3oJvKCjJZL/rlryyNwVeJH5OSihHret+dXoxSQWVhnDQuu17iQkyUIYRTsfFTqppAmQIfdq2VIKgOsimGcb4xCo9HMXKHmnwVP29kYHQeiRCOynADPS8NxH/89qpiS6DjMkkNVSS2UNRyrGJ8aQQ3GOKEmPz9xgQxexfMRmAAmJsbUVbgj8feZE0KmX/vHx2VylVK3kdBXSEjtEp8tEFqqJrVEN1RNAjekav6M15cl6cd+djNrrk5DsH6A+czx+xXpeC</latexit> Machine Learning <latexit sha1_base64="UmHDp/+ZbhkGu2OLzWSXl+Ek3fo=">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</latexit> inf " + 1 n Pn i=1 si s.t. ✓ 2 ⇥, 2 R +, si 2 R, ⇣ij 2 Rd ( `j)⇤2(✓, ⇣ij) + c⇤1(⇣ij/ , zi)  si <latexit sha1_base64="mRyHk57TAlR7oNv/yK+Z34/82w0=">AAACG3icbVC7SgNBFJ31GeMramkzGARtlt0gaimksVQxKiQhzE7umsF5LDN3g2HJf9j4KzYWilgJFv6Nk5jC14GBwzn3MPeeJJPCYRR9BFPTM7Nz86WF8uLS8spqZW39wpnccmhwI429SpgDKTQ0UKCEq8wCU4mEy+SmPvIv+2CdMPocBxm0FbvWIhWcoZc6lVrLpSlTQg5oC+EWk7Q474GxoIZ0p250H27pGaTGqlyOI7thp1KNwmgM+pfEE1IlE5x0Km+truG5Ao1cMueacZRhu2AWBZcwLLdyBxnjN+wamp5qpsC1i/FtQ7rtlS71C/inkY7V74mCKecGKvGTimHP/fZG4n9eM8f0sF0IneUImn99lOaSoqGjomhXWODoe+kKxq3wu1LeY5Zx9HWWfQnx75P/kotaGO+He6e16lE4qaNENskW2SExOSBH5JickAbh5I48kCfyHNwHj8FL8Po1OhVMMhvkB4L3T6W8obc=</latexit> Theorem (Convex Reformulation). <latexit sha1_base64="j2bnJI7xybZMZXSD/oFTMJMmQqE=">AAACGnicbVBNSwMxEM36WetX1aOXYCso6LJbRD0WvHhUsCp0S8mmUxvNZpdktliX/R1e/CtePCjiTbz4b0xrD2p9EPJ4b2YyeWEihUHP+3QmJqemZ2YLc8X5hcWl5dLK6rmJU82hzmMZ68uQGZBCQR0FSrhMNLAolHAR3hwN/IseaCNidYb9BJoRu1KiIzhDK7VKfiUAKVvZdb6VBcNxmYZ2HmAXkOU7d9sVymPVg9tde3HWA7dVKnuuNwQdJ/6IlMkIJ63Se9COeRqBQi6ZMQ3fS7CZMY2CS8iLQWogYfyGXUHDUsUiMM1suEtON63Spp1Y26OQDtWfHRmLjOlHoa2MGHbNX28g/uc1UuwcNjOhkhRB8e+HOqmkGNNBTrQtNHCUfUsY18LuSnmXacbRplm0Ifh/vzxOzquuv+/unVbLNXcUR4Gskw2yRXxyQGrkmJyQOuHknjySZ/LiPDhPzqvz9l064Yx61sgvOB9fm0qhJg==</latexit> `j(✓, z) convex-concave. <latexit sha1_base64="7AZ6cDyCtvBqnQbYZSjKAitplV8=">AAACKXicbVBNSwMxFMz6bf2qevQSbIUKsuyKqMeCF48K1ha6pbxN32owm12SrFiX/h0v/hUvCop69Y+YbntQ60BgmJmXvEyYCq6N5304U9Mzs3PzC4ulpeWV1bXy+salTjLFsMESkahWCBoFl9gw3AhspQohDgU2w5uTod+8RaV5Ii9MP8VODFeSR5yBsVK3XA90FEHMRZ/WON+lVVYL7tHAHs2D4vac9UEO7ge7Vco1ZYm8xTvKJa0WsarbLVc81ytAJ4k/JhUyxlm3/BL0EpbFKA0ToHXb91LTyUEZzgQOSkGmMQV2A1fYtlRCjLqTF7sM6I5VejRKlD3S0EL9OZFDrHU/Dm0yBnOt/3pD8T+vnZnouJNzmWYGJRs9FGWCmoQOa6M9rpAZ21KPA1Pc7krZNShgxpZbsiX4f788SS73Xf/QPTjfr9TdcR0LZItskxrxyRGpk1NyRhqEkQfyRF7Jm/PoPDvvzucoOuWMZzbJLzhf32TQpWw=</latexit> (ii) c(⇣, z) is convex in ⇣. <latexit sha1_base64="RQ4dKCJyPSjV5DgP+Bturl+qiM0=">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</latexit> (i) `(✓, z) = maxj2[J] `j(✓, z), <latexit sha1_base64="WpvjGo+tV0Gf3DlL1Yf+30vHoZA=">AAACC3icbVA7T8MwGHTKq5RXgJHFaoXEFCUVAsYiFsYi0YfURJXjOK1V24lsB6mKurPwV1gYQIiVP8DGv8FpM0DLSZZOd/fZ/i5MGVXadb+tytr6xuZWdbu2s7u3f2AfHnVVkklMOjhhieyHSBFGBeloqhnpp5IgHjLSCyc3hd97IFLRRNzraUoCjkaCxhQjbaShXfdVHCNO2RT6mYiILC7Kr5XKeFok1MwZ2g3XceeAq8QrSQOUaA/tLz9KcMaJ0JghpQaem+ogR1JTzMis5meKpAhP0IgMDBWIExXk811m8NQoEYwTaY7QcK7+nsgRV2rKQ5PkSI/VsleI/3mDTMdXQU5Fmmki8OKhOGNQJ7AoBkZUEqxNDxFFWFLzV4jHSCKsTX01U4K3vPIq6TYd78I5v2s2Wk5ZRxWcgDo4Ax64BC1wC9qgAzB4BM/gFbxZT9aL9W59LKIVq5w5Bn9gff4ALX2btw==</latexit> Assumptions. <latexit sha1_base64="VjOZLMgi4zKoM3rOmnJQgkn0NZc=">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</latexit> (i’) `(✓, z) = `(✓>z). <latexit sha1_base64="vU7aTTzpLB+ht13FjmVWje+PtOA=">AAACAXicbVBLSwMxGMz6rPW16kXwEiyCp7JbRD0WvHisYB/QXUo2m21D81iSrFCWevGvePGgiFf/hTf/jdl2D9o6EBhmvi/JTJQyqo3nfTsrq2vrG5uVrer2zu7evntw2NEyU5i0sWRS9SKkCaOCtA01jPRSRRCPGOlG45vC7z4QpakU92aSkpCjoaAJxchYaeAeBzpJEKdsAoNMxEQVF+VSTQduzat7M8Bl4pekBkq0Bu5XEEuccSIMZkjrvu+lJsyRMhQzMq0GmSYpwmM0JH1LBeJEh/kswRSeWSWGiVT2CANn6u+NHHGtJzyykxyZkV70CvE/r5+Z5DrMqUgzQwSeP5RkDBoJizpgTBXBxqaPKcKK2r9CPEIKYWNLq9oS/MXIy6TTqPuX9Yu7Rq1ZL+uogBNwCs6BD65AE9yCFmgDDB7BM3gFb86T8+K8Ox/z0RWn3DkCf+B8/gBI/5dg</latexit> or <latexit sha1_base64="7zV5f8g/OYdJ674qK+e9j2D49ng=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0hE1GPBi8eK9gPaUDbbTbt0swm7E6GE/gQvHhTx6i/y5r9x2+agrQ8GHu/NMDMvTKUw6HnfTmltfWNzq7xd2dnd2z+oHh61TJJpxpsskYnuhNRwKRRvokDJO6nmNA4lb4fj25nffuLaiEQ94iTlQUyHSkSCUbTSQw+TfrXmud4cZJX4BalBgUa/+tUbJCyLuUImqTFd30sxyKlGwSSfVnqZ4SllYzrkXUsVjbkJ8vmpU3JmlQGJEm1LIZmrvydyGhsziUPbGVMcmWVvJv7ndTOMboJcqDRDrthiUZRJggmZ/U0GQnOGcmIJZVrYWwkbUU0Z2nQqNgR/+eVV0rpw/Sv38v6yVneLOMpwAqdwDj5cQx3uoAFNYDCEZ3iFN0c6L86787FoLTnFzDH8gfP5A1vTjcw=</latexit> ! <latexit sha1_base64="rhTrN537fh12yv9VtBq3L9tVut0=">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</latexit> B|·| " (P 1) = {Q : E Z⇠Q[|Z|]  "} <latexit sha1_base64="S/yan/wPgrTIgXVt5i7qO7ll2Jg=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXIWkiLosdONGqGAf0IQymUzaoZMHMzdiiV34K25cKOLW33Dn3zhts9DWAxcO59zLvff4qeAKbPvbKK2srq1vlDcrW9s7u3vm/kFbJZmkrEUTkciuTxQTPGYt4CBYN5WMRL5gHX/UmPqdeyYVT+I7GKfMi8gg5iGnBLTUN49cFYYk4mKMXWAP4If5TbMx6ZtV27JnwMvEKUgVFWj2zS83SGgWsRioIEr1HDsFLycSOBVsUnEzxVJCR2TAeprGJGLKy2f3T/CpVgIcJlJXDHim/p7ISaTUOPJ1Z0RgqBa9qfif18sgvPJyHqcZsJjOF4WZwJDgaRg44JJR0L8HnFDJ9a2YDokkFHRkFR2Cs/jyMmnXLOfCOr+tVetWEUcZHaMTdIYcdInq6Bo1UQtR9Iie0St6M56MF+Pd+Ji3loxi5hD9gfH5AxAYlhI=</latexit> MPC <latexit sha1_base64="1ovEWTPSSLGKApYMDnR08fZFNkE=">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</latexit> Wc(Q, P n) = inf 2 (Q,P n) Z Rd⇥Rd c(Z1, Z2)d (Z1, Z2) <latexit sha1_base64="YftGhE2/DuTEYvhmpkwrkq7Y6Lc=">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</latexit> Q? is discrete with finite support. <latexit sha1_base64="42a9BDbYHb/D+aturzmWj92WO9M=">AAACA3icbVDLSgMxFM34rPVVdaebYBFcDTOlqOCm4MZlBfuAzlAymUwbmseQZIQyFNz4K25cKOLWn3Dn35i2s9DWA4HDOfdyc06UMqqN5307K6tr6xubpa3y9s7u3n7l4LCtZaYwaWHJpOpGSBNGBWkZahjppoogHjHSiUY3U7/zQJSmUtybcUpCjgaCJhQjY6V+5TjQSYI4ZWPou8E1bCqZokFhVj3XmwEuE78gVVCg2a98BbHEGSfCYIa07vleasIcKUMxI5NykGmSIjxCA9KzVCBOdJjPMkzgmVVimEhlnzBwpv7eyBHXeswjO8mRGepFbyr+5/Uyk1yFORVpZojA80NJxqCRcFoIjKki2Nj8MUVYUftXiIdIIWxsbWVbgr8YeZm0a65/4dbvatVGvaijBE7AKTgHPrgEDXALmqAFMHgEz+AVvDlPzovz7nzMR1ecYucI/IHz+QMIVpcZ</latexit> 1. Propagation <latexit sha1_base64="J8aiEfOOC9xYhqux+0RDSt8iUe4=">AAACA3icbVDLSgMxFM3UV62vqjvdBIvgapgpRQU3hW5cVrAP6Awlk8m0oXkMSUYopeDGX3HjQhG3/oQ7/8a0nYW2HggczrmXm3OilFFtPO/bKaytb2xuFbdLO7t7+wflw6O2lpnCpIUlk6obIU0YFaRlqGGkmyqCeMRIJxo1Zn7ngShNpbg345SEHA0ETShGxkr98kmgkwRxysaw6gY3sCF5mpncrHiuNwdcJX5OKiBHs1/+CmKJM06EwQxp3fO91IQTpAzFjExLQaZJivAIDUjPUoE40eFknmEKz60Sw0Qq+4SBc/X3xgRxrcc8spMcmaFe9mbif14vM8l1OKHCxiICLw4lGYNGwlkhMKaKYGPzxxRhRe1fIR4ihbCxtZVsCf5y5FXSrrr+pVu7q1bqtbyOIjgFZ+AC+OAK1MEtaIIWwOARPINX8OY8OS/Ou/OxGC04+c4x+APn8wcgsZcp</latexit> 2. Computation Liviu Aolaritei + Nicolas Lanzetti, 
 Marta Fochesato, 
 Soroosh Shafiee, 
 Antonio Terpin, 
 Daniel Kuhn, 
 John Lygeros, … 6 0. Capturing 3. Applications: 
 today: control
  7. Propagation of OT Ambiguity Sets <latexit sha1_base64="QIl416K627bV6D5dHLh1RrhcQzI=">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</latexit> f(Bc " (P

    n)) = ? <latexit sha1_base64="/+FRpYVTL/Q5DmLt2ZGqfvDazEI=">AAACAnicbVDLSsNAFL3xWesr6krcDBbBVUmkqMuCG5cV7AOaUCbTaTt0MgkzEyGE4sZfceNCEbd+hTv/xkmahbYeGDicc19zgpgzpR3n21pZXVvf2KxsVbd3dvf27YPDjooSSWibRDySvQArypmgbc00p71YUhwGnHaD6U3udx+oVCwS9zqNqR/isWAjRrA20sA+zrxiSEZSLGZeiPUkCFBrIGYDu+bUnQJombglqUGJ1sD+8oYRSUIqNOFYqb7rxNrPsNSMcDqreomiMSZTPKZ9QwUOqfKzYvsMnRlliEaRNE9oVKi/OzIcKpWGganMb1SLXi7+5/UTPbr2MybiRFNB5otGCUc6QnkeaMgkJZqnhmAimbkVkQmWmGiTWtWE4C5+eZl0LuruZb1x16g1G2UcFTiBUzgHF66gCbfQgjYQeIRneIU368l6sd6tj3npilX2HMEfWJ8/8cqXvQ==</latexit> P n <latexit sha1_base64="k1AtdIimUWBqygfvFGKKOZEHPFk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE8lUSKeix48VjBfkAaymY7aZdudsPupFBCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXpYIb9LxvZ2Nza3tnt7RX3j84PDqunJy2jco0gxZTQuluRA0ILqGFHAV0Uw00iQR0ovH93O9MQBuu5BNOUwgTOpQ85oyilYLehGpIDRdK9itVr+Yt4K4TvyBVUqDZr3z1BoplCUhkghoT+F6KYU41ciZgVu5lBlLKxnQIgaWSJmDCfHHyzL20ysCNlbYl0V2ovydymhgzTSLbmVAcmVVvLv7nBRnGd2HOZZohSLZcFGfCReXO/3cHXANDMbWEMs3trS4bUU0Z2pTKNgR/9eV10r6u+Te1+mO92qgXcZTIObkgV8Qnt6RBHkiTtAgjijyTV/LmoPPivDsfy9YNp5g5I3/gfP4AuIeRgg==</latexit> " <latexit sha1_base64="GUD9jfdS/NnObjoPykcSdja/fbk=">AAACIHicbVDNS8MwHE39nPNr6tFLcAjzMloZTgRh6MXjBPcBay1plm1haVqSdFBK/xQv/itePCiiN/1rTLcKuvkg8Hjv/ZJfnhcyKpVpfhpLyyura+uFjeLm1vbObmlvvy2DSGDSwgELRNdDkjDKSUtRxUg3FAT5HiMdb3yd+Z0JEZIG/E7FIXF8NOR0QDFSWnJLddtHauR58MpN7AkSJJSUBTy9x5XEnl6f4Bjx9CfWdHl6Ai8u3VLZrJpTwEVi5aQMcjTd0ofdD3DkE64wQ1L2LDNUToKEopiRtGhHkoQIj9GQ9DTlyCfSSaYbpPBYK304CIQ+XMGp+nsiQb6Use/pZLannPcy8T+vF6nBuZNQHkaKcDx7aBAxqAKYtQX7VBCsWKwJwoLqXSEeIYGw0p0WdQnW/JcXSfu0ap1Va7e1cqOW11EAh+AIVIAF6qABbkATtAAGD+AJvIBX49F4Nt6M91l0ychnDsAfGF/fd8mjtA==</latexit> Bc " (P n) := <latexit sha1_base64="j+GoXDcncB2rRFn7R42mCXd9rFE=">AAACIXicbVDLSgMxFM34rPU16tJNsAh1U2akaJdFNy4r2Ad06pBJM21oJhmSTKEM8ytu/BU3LhTpTvwZ02kFbT0QOJxzL+fmBDGjSjvOp7W2vrG5tV3YKe7u7R8c2kfHLSUSiUkTCyZkJ0CKMMpJU1PNSCeWBEUBI+1gdDvz22MiFRX8QU9i0ovQgNOQYqSN5Nu10PdKXoT0MAjgjZ96YyRJrCgTPHvE5dTLI1I8QTz7GWv4PLvw7ZJTcXLAVeIuSAks0PDtqdcXOIkI15ghpbquE+teiqSmmJGs6CWKxAiP0IB0DeUoIqqX5vEZPDdKH4ZCmsc1zNXfGymKlJpEgZmcHamWvZn4n9dNdFjrpZTHiSYcz4PChEEt4Kwu2KeSYM0mhiAsqbkV4iGSCGtTatGU4C5/eZW0LivuVaV6Xy3Vq4s6CuAUnIEycME1qIM70ABNgMETeAFv4N16tl6tD2s6H12zFjsn4A+sr2+qKqRr</latexit> f# Bc " (P n) <latexit sha1_base64="DWUYnMd3ecJU3bhhZjYOsArTp3g=">AAACBXicbVDNS8MwHE39nPOr6lEPwSHMy2hlqMeBF48T3AespaRZuoWlaUlSoZRevPivePGgiFf/B2/+N6ZdD7r5IPB47/eV58eMSmVZ38bK6tr6xmZtq769s7u3bx4c9mWUCEx6OGKRGPpIEkY56SmqGBnGgqDQZ2Tgz24Kf/BAhKQRv1dpTNwQTTgNKEZKS555EjQzpxyT4RTx3AmRmvo+7Ho8P/fMhtWySsBlYlekASp0PfPLGUc4CQlXmCEpR7YVKzdDQlHMSF53EklihGdoQkaachQS6Wbl+hyeaWUMg0joxxUs1d8dGQqlTENfVxZHykWvEP/zRokKrt2M8jhRhOP5oiBhUEWwiASOqSBYsVQThAXVt0I8RQJhpYOr6xDsxS8vk/5Fy75ste/ajU67iqMGjsEpaAIbXIEOuAVd0AMYPIJn8ArejCfjxXg3PualK0bVcwT+wPj8AY/SmJI=</latexit> f(P n) <latexit sha1_base64="9dVNL0wEdrlm5eP+Act+ZB+nFrY=">AAACFHicbVDLSgMxFM34rPU16tJNsAiCUmZKUZcFN7qrYB/QlpJJ77ShmcyQ3FFK6Ue48VfcuFDErQt3/o3pY6GtB0IO59xLck6QSGHQ876dpeWV1bX1zEZ2c2t7Z9fd26+aONUcKjyWsa4HzIAUCiooUEI90cCiQEIt6F+N/do9aCNidYeDBFoR6yoRCs7QSm33tGnCkEVCDuhNSFWMZ5QzRR/AXgmmGqhAiqLbQzlouzkv701AF4k/IzkyQ7ntfjU7MU8jUMglM6bhewm2hkyj4BJG2WZqIGG8z7rQsFSxCExrOAk1osdW6dAw1vYopBP198aQRcYMosBORgx7Zt4bi/95jRTDy9ZQqCRFUHz6UJhKijEdN0Q7QgO3cS1hXAv7V8p7TDOOtsesLcGfj7xIqoW8f54v3hZypeKsjgw5JEfkhPjkgpTINSmTCuHkkTyTV/LmPDkvzrvzMR1dcmY7B+QPnM8fu8ad+Q==</latexit> If not, can we capture it tightly <latexit sha1_base64="PTE5ybiHcmaFobRg5/0PCJ2XthM=">AAACEXicbVC7SgNBFJ2NrxhfUUubwSCkCrshqJ0BGzsj5AVJCLOTu8mQ2dll5q4SQn7Bxl+xsVDE1s7Ov3HyKDTxwIXDOffO3Hv8WAqDrvvtpNbWNza30tuZnd29/YPs4VHdRInmUOORjHTTZwakUFBDgRKasQYW+hIa/vB66jfuQRsRqSqOYuiErK9EIDhDK3Wz+bYJAhYKOaIPAgeUqQgHoOltldpHRD8ROKIG8KqbzbkFdwa6SrwFyZEFKt3sV7sX8SQEhVwyY1qeG2NnzDQKLmGSaScGYsaHrA8tSxULwXTGs4sm9MwqPRpE2pZCOlN/T4xZaMwo9G1nyHBglr2p+J/XSjC47IyFihMExecfBYmkGNFpPLQnNHC0afQE41rYXSkfMM042hAzNgRv+eRVUi8WvPNC6a6YK5cWcaTJCTkleeKRC1ImN6RCaoSTR/JMXsmb8+S8OO/Ox7w15SxmjskfOJ8/LzedMQ==</latexit> with another OT ambiguity set? <latexit sha1_base64="5Hu0I4bDpU5j1ARUBUO821O8xxE=">AAACD3icbVDLSgMxFM3UV62vUZdugkVxVWZKUXcW3OjKCn1BW0omzbShSWZI7gil9A/c+CtuXCji1q07/8a0nYW2HrhwOOfe5N4TxIIb8LxvJ7Oyura+kd3MbW3v7O65+wd1EyWashqNRKSbATFMcMVqwEGwZqwZkYFgjWB4PfUbD0wbHqkqjGLWkaSveMgpASt13dO2CUMiuRjhW4NhwA0mCt9VsX2C9xMOI2wYXHXdvFfwZsDLxE9JHqWodN2vdi+iiWQKqCDGtHwvhs6YaOBUsEmunRgWEzokfdayVBHJTGc8u2eCT6zSw2GkbSnAM/X3xJhIY0YysJ2SwMAselPxP6+VQHjZGXMVJ8AUnX8UJgJDhKfh4B7XjILNoscJ1dzuiumAaELBRpizIfiLJy+TerHgnxdK98V8uZTGkUVH6BidIR9doDK6QRVUQxQ9omf0it6cJ+fFeXc+5q0ZJ505RH/gfP4Axqeb0w==</latexit> Is this an OT ambiguity set? <latexit sha1_base64="gzBBd/K4jjBYVFimyExo36VKlsI=">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</latexit> P n = 1 n n X i=1 zi <latexit sha1_base64="swE2QqOkcbjnYykzdSPwqYs1FkI=">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</latexit> f# P n = 1 n n X i=1 f(zi) <latexit sha1_base64="mfesxccVPkCafw+3WOejL7GtLKM=">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</latexit> Pushforward operation: Z ⇠ P f(Z) ⇠ f# P 8 <latexit sha1_base64="RbpwAWnlRlKzN0bhuN7LCT3y/DE=">AAACC3icbVDLSgMxFM3UV62vqks3oUWomzJTirosiOCygn1AW0omc6cNTTJDkhFL6d6Nv+LGhSJu/QF3/o1pOwttPRA4nHMfucePOdPGdb+dzNr6xuZWdju3s7u3f5A/PGrqKFEUGjTikWr7RANnEhqGGQ7tWAERPoeWP7qa+a17UJpF8s6MY+gJMpAsZJQYK/Xzha4OQyIYH+PrByJiDrgUgAElmLTbGT0r9/NFt+zOgVeJl5IiSlHv57+6QUQTAdJQTrTueG5sehOi7DgO01w30RATOiID6FgqiQDdm8xvmeJTqwQ4jJR90uC5+rtjQoTWY+HbSkHMUC97M/E/r5OY8LI3YTJODEi6WBQmHJsIz4LBAVNAjc0hYIQqZv+K6ZAoQm0YOmdD8JZPXiXNStk7L1dvK8VaNY0ji05QAZWQhy5QDd2gOmogih7RM3pFb86T8+K8Ox+L0oyT9hyjP3A+fwCJKJqu</latexit> Example (deterministic). <latexit sha1_base64="ooshO5wvzCj+W8mCwRnm5f5cVz0=">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</latexit> Bc " (z0) := {z 2 Rd : c(z z0)  "} <latexit sha1_base64="rIs8EByG696cY73fF1dvYxW85rk=">AAACIHicbVDLSsNAFJ34rPUVdelmsBVcSElKseKq4MZlFfuAJpbJZNKOnUnCzKRQQj/Fjb/ixoUiutOvcdJ2UVsPDJw5917uPceLGZXKsr6NldW19Y3N3FZ+e2d3b988OGzKKBGYNHDEItH2kCSMhqShqGKkHQuCuMdIyxtcZ/XWkAhJo/BejWLictQLaUAxUlrqmlVHBgHilI1gMYBX0OFI9T0P3j340FHR/L/onEOPPhKs6JB0zYJVsiaAy8SekQKYod41vxw/wgknocIMSdmxrVi5KRKKYkbGeSeRJEZ4gHqko2mIOJFuOjE4hqda8WEQCf1CBSfq/ESKuJQj7unO7F65WMvE/2qdRAWXbkrDOFEkxNNFQcKgdp6lBX0qtF8djk8RFlTfCnEfCYSVzjSvQ7AXLS+TZrlkX5Qqt+VCrTKLIweOwQk4Azaoghq4AXXQABg8gRfwBt6NZ+PV+DA+p60rxmzmCPyB8fMLB5ChoQ==</latexit> f : Rd ! Rd bijective <latexit sha1_base64="J3r+J0HAvI2EqMt1jC26nsdjyo0=">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</latexit> f(Bc " (z0)) = {f(z) 2 Rd : c(z z0)  "}] or <latexit sha1_base64="VGj9o+zmHQnONUnR0anfRijtTV4=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkqMeCF48V7Qe0oWy2k3bpZhN2N0IN/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4Zua3H1FpHssHM0nQj+hQ8pAzaqx0/9R3++WKW3XnIKvEy0kFcjT65a/eIGZphNIwQbXuem5i/Iwqw5nAaamXakwoG9Mhdi2VNELtZ/NTp+TMKgMSxsqWNGSu/p7IaKT1JApsZ0TNSC97M/E/r5ua8NrPuExSg5ItFoWpICYms7/JgCtkRkwsoUxxeythI6ooMzadkg3BW355lbQuqt5ltXZXq9RreRxFOIFTOAcPrqAOt9CAJjAYwjO8wpsjnBfn3flYtBacfOYY/sD5/AEL8I2c</latexit> z0 <latexit sha1_base64="3RoHHu675qYqLgNkuwEbW/QbGPA=">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</latexit> ) <latexit sha1_base64="9NooNlx4ru8c8x9IHlW+K8rEjA0=">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</latexit> f(Bc " (z0)) = ˜ z 2 Rd : c(f 1(˜ z) f 1(f(z0)))  " = Bc f 1 " (f(z0)) or <latexit sha1_base64="9NooNlx4ru8c8x9IHlW+K8rEjA0=">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</latexit> f(Bc " (z0)) = ˜ z 2 Rd : c(f 1(˜ z) f 1(f(z0)))  " = Bc f 1 " (f(z0))
  8. Pushforward of OT Ambiguity Sets <latexit sha1_base64="IxrbsmMSaxGWOCxYuHQbUdD7Ju8=">AAACBHicbVA9SwNBEN2LXzF+RS3TLAbBKtyFoJYBG8sI+YIkhL3NXLJk9+7YnRPDkcLGv2JjoYitP8LOf+Pmo9DEBwOP92aYmefHUhh03W8ns7G5tb2T3c3t7R8cHuWPT5omSjSHBo9kpNs+MyBFCA0UKKEda2DKl9Dyxzczv3UP2ogorOMkhp5iw1AEgjO0Uj9f6JogYErICe0iPKAfpPURRBpUadrPF92SOwddJ96SFMkStX7+qzuIeKIgRC6ZMR3PjbGXMo2CS5jmuomBmPExG0LH0pApML10/sSUnltlQINI2wqRztXfEylTxkyUbzsVw5FZ9Wbif14nweC6l4owThBCvlgUJJJiRGeJ0IHQwNEGMBCMa2FvpXzENONoc8vZELzVl9dJs1zyLkuVu3KxWlnGkSUFckYuiEeuSJXckhppEE4eyTN5JW/Ok/PivDsfi9aMs5w5JX/gfP4AN/yYbA==</latexit> Theorem. <latexit sha1_base64="RwpP441Urz70KBxr6ENnGKKWbcs=">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</latexit>

    f# Bc " (P) = Bc f 1 " (f# P) <latexit sha1_base64="pI7xg9MFWzplSwLNCQCV3ObE/KU=">AAACEXicbVC7SgNBFJ2NrxhfUUubwURIIctuCCrYBGy0i2AekF3C7ORuMmb2wcxsICz5BRt/xcZCEVs7O//GSbKFRg8MHM45lzv3eDFnUlnWl5FbWV1b38hvFra2d3b3ivsHLRklgkKTRjwSHY9I4CyEpmKKQycWQAKPQ9sbXc389hiEZFF4pyYxuAEZhMxnlCgt9YoVR/o+CRifYNt0LvGNj8t+GTOJPXYPVLExnGI1BB0tWaY1B/5L7IyUUIZGr/jp9COaBBAqyomUXduKlZsSoRjlMC04iYSY0BEZQFfTkAQg3XR+0RSfaKWP/UjoFyo8V39OpCSQchJ4OhkQNZTL3kz8z+smyr9wUxbGiYKQLhb5CccqwrN6cJ8JfbVuo88IFUz/FdMhEYQqXWJBl2Avn/yXtKqmfWbWbqulei2rI4+O0DGqIBudozq6Rg3URBQ9oCf0gl6NR+PZeDPeF9Gckc0col8wPr4BU0Kbag==</latexit> 1. If f is bijective, then 9 <latexit sha1_base64="z34ehPfepVh2DMe2TMc+FasI/bw=">AAACDnicbVDLTgIxFO3gC/GFunTTCCS4ITOEqEsSNy4xkUfCTMidUqCh05m0HZPJhC9w46+4caExbl27828sAwsFT9Lk5Jx729PjR5wpbdvfVm5jc2t7J79b2Ns/ODwqHp90VBhLQtsk5KHs+aAoZ4K2NdOc9iJJIfA57frTm7nffaBSsVDc6ySiXgBjwUaMgDbSoFipllM3uyYlCYiZG4Ce+D5uzcoYpM+0BJlcDIolu2ZnwOvEWZISWqI1KH65w5DEARWacFCq79iR9lKQmhFOZwU3VjQCMoUx7RsqIKDKS7McM1wxyhCPQmmO0DhTf2+kECiVBL6ZnKdVq95c/M/rx3p07aVMRLGmgiweGsUc6xDPu8FDJinRPDEEiGQmKyYTkEC0abBgSnBWv7xOOvWac1lr3NVLzcayjjw6Q+eoihx0hZroFrVQGxH0iJ7RK3qznqwX6936WIzmrOXOKfoD6/MH1mmb7w==</latexit> (P arbitrary) <latexit sha1_base64="XK98Ks0MwQv2wQNL+lUFzXSE9fU=">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</latexit> f# Bc " (P) ✓ Bc f 1 " (f# P) <latexit sha1_base64="QI4/ilt3jUVCIvPAd2x4Liqnzqc=">AAAECnicdVLLbtNAFJ3WPEp4tIUlmxFpJRZViIN4SGzKQwh2RZC2UmxF4/F1M3QeZmacNoz8B/wAv8EOseiGBXwCf8M4cRFO0iuNfHTuuU/fJOfM2G73z8pqcOnylatr11rXb9y8tb6xeXvfqEJT6FPFlT5MiAHOJPQtsxwOcw1EJBwOkuOXlf9gDNowJT/YSQ6xIEeSZYwS66nhxsPIZBkRjE9wrxM9w28zvJVtYWYwkx+BWjaGB6bQNdzBdgQ+qt3tdKeGF0FYgzaqbW+4uXoWpYoWAqSlnBgzCLu5jR3RllEOZSsqDOSEHpMjGHgoiQCzk45ZbqYwdtNBS7ztnSnOlPZPWjxl/w92RBgzEYlXCmJHZt5Xkct8g8JmT2PHZF5YkHRWKCs4tgpXW8Mp034DfkkpI1Qz3zamI6IJtX63jSonxEx8C42ZXFXQKsVNubyj5TM0aWo+FcrCYopqFWaxnjbZEjZV89p0mqDJnWZ+tLLV2sZjP7aqRnwF/s9peO87U/y1j3CJB2np+uU5EqWT5RLlc56PSALWRVUHtXj2aUUSTqgSgsjURYaJnMNpOQhj59NwS4auHZZzqqqlmeRfugtUSivpF+a1g3jGuLC8KKXSn0Grprp7rvYnH84f+CLY73XCx51H73rt3Rf18a+hu+geuo9C9ATtojdoD/URRV/RGfqFfgdfgm/B9+DHTLq6UsfcQQ0Lfv4FG0xizA==</latexit> 2. If f is injective/surjective, then <latexit sha1_base64="ElaIVyLanfUbjio1BWp5qeOs+no=">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</latexit> A# Bc " (P) ✓ Bc A† " (A# P) <latexit sha1_base64="P3EmlpEkwSbmmQU4aoSnzP24cag=">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</latexit> 3. If f(z) = Az with a general matrix A, then <latexit sha1_base64="tLJqkzGnpyD7buS5ynHR89GuWY0=">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</latexit> with equality ⌘ if A is full row-rank <latexit sha1_base64="11qSHs/RiI0ehFHlhP+stUC5GsY=">AAACDHicbVDLSgMxFM3UV62vqks3wSLUTZkpRV0W3LisYFuhHUomvdOGJpMhyQjD0A9w46+4caGIWz/AnX9j2s5CWw8EDufce3PvCWLOtHHdb6ewtr6xuVXcLu3s7u0flA+POlomikKbSi7VfUA0cBZB2zDD4T5WQETAoRtMrmd+9wGUZjK6M2kMviCjiIWMEmOlQbnS12FIBOMprrIIEzxkYQgKIoNHIAUYlZ7bKrfmzoFXiZeTCsrRGpS/+kNJE2GnUE607nlubPyMKMMoh2mpn2iICZ2QEfQsjYgA7WfzY6b4zCpDHEpln91irv7uyIjQOhWBrRTEjPWyNxP/83qJCa/8jEVxYiCii4/ChGMj8SwZe7kCamwQQ0aoYnZXTMdEEWpsfiUbgrd88irp1GveRa1xW680G3kcRXSCTlEVeegSNdENaqE2ougRPaNX9OY8OS/Ou/OxKC04ec8x+gPn8wf16Jrl</latexit> (in a di↵erent geometry) <latexit sha1_base64="/qCZW2bqqU1GCODuHYQC5cKD1dk=">AAACBHicbVC7SgNBFJ2Nrxhfq5ZpBoNgFXZDUMuAjWUE84AkhLuTu8mQmd1lZlZYQgobf8XGQhFbP8LOv3HyKDTxwMDhnHu5c06QCK6N5307uY3Nre2d/G5hb//g8Mg9PmnqOFUMGywWsWoHoFHwCBuGG4HtRCHIQGArGN/M/NYDKs3j6N5kCfYkDCMecgbGSn232NVhCJKLjCLojJqYJipOYAgG+27JK3tz0HXiL0mJLFHvu1/dQcxSiZFhArTu+F5iehNQhjOB00I31ZgAG8MQO5ZGIFH3JvMQU3pulQENY2VfZOhc/b0xAal1JgM7KcGM9Ko3E//zOqkJr3sTHiWpwYgtDoWpmEWdNUIHXCEztoABB6a4/StlI1DAjO2tYEvwVyOvk2al7F+Wq3eVUq26rCNPiuSMXBCfXJEauSV10iCMPJJn8krenCfnxXl3PhajOWe5c0r+wPn8AcHZmCE=</latexit> easy to propagate <latexit sha1_base64="BFPuY7kYqpHlr3hbCsPYBWBbCmA=">AAACEXicbVC7SgNBFJ2Nrxhfq5Y2g0GITdgNQS0DNpYRTCIkIczO3k2GzMwuM7NCCPkFG3/FxkIRWzs7/8bJZgtNPHDhcO6Lc4KEM20879sprK1vbG4Vt0s7u3v7B+7hUVvHqaLQojGP1X1ANHAmoWWY4XCfKCAi4NAJxtfzfucBlGaxvDOTBPqCDCWLGCXGSgO30tNRRATjE1whGjOJzQhwCAaUYNL+ZxRTe/984Ja9qpcBrxI/J2WUozlwv3phTFMB0lBOtO76XmL6U6LsSQ6zUi/VkBA6JkPoWiqJAN2fZo5m+MwqIY5iZUsanKm/N6ZEaD0RgZ0UxIz0cm8u/tfrpia66k+ZTFIDki4eRSnHJsbzeHDIFFBj0wgZoYpl9kdEEWoD0SUbgr9seZW0a1X/olq/rZUb9TyOIjpBp6iCfHSJGugGNVELUfSIntErenOenBfn3flYjBacfOcY/YHz+QOd7JzY</latexit> (as in the deterministic case) closure: is an OT 
 ambiguity set ! <latexit sha1_base64="k1AtdIimUWBqygfvFGKKOZEHPFk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE8lUSKeix48VjBfkAaymY7aZdudsPupFBCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXpYIb9LxvZ2Nza3tnt7RX3j84PDqunJy2jco0gxZTQuluRA0ILqGFHAV0Uw00iQR0ovH93O9MQBuu5BNOUwgTOpQ85oyilYLehGpIDRdK9itVr+Yt4K4TvyBVUqDZr3z1BoplCUhkghoT+F6KYU41ciZgVu5lBlLKxnQIgaWSJmDCfHHyzL20ysCNlbYl0V2ovydymhgzTSLbmVAcmVVvLv7nBRnGd2HOZZohSLZcFGfCReXO/3cHXANDMbWEMs3trS4bUU0Z2pTKNgR/9eV10r6u+Te1+mO92qgXcZTIObkgV8Qnt6RBHkiTtAgjijyTV/LmoPPivDsfy9YNp5g5I3/gfP4AuIeRgg==</latexit> " <latexit sha1_base64="k1AtdIimUWBqygfvFGKKOZEHPFk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE8lUSKeix48VjBfkAaymY7aZdudsPupFBCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXpYIb9LxvZ2Nza3tnt7RX3j84PDqunJy2jco0gxZTQuluRA0ILqGFHAV0Uw00iQR0ovH93O9MQBuu5BNOUwgTOpQ85oyilYLehGpIDRdK9itVr+Yt4K4TvyBVUqDZr3z1BoplCUhkghoT+F6KYU41ciZgVu5lBlLKxnQIgaWSJmDCfHHyzL20ysCNlbYl0V2ovydymhgzTSLbmVAcmVVvLv7nBRnGd2HOZZohSLZcFGfCReXO/3cHXANDMbWEMs3trS4bUU0Z2pTKNgR/9eV10r6u+Te1+mO92qgXcZTIObkgV8Qnt6RBHkiTtAgjijyTV/LmoPPivDsfy9YNp5g5I3/gfP4AuIeRgg==</latexit> " <latexit sha1_base64="qZLECDfqk/Xe4H8vb+99VR+0LTI=">AAAEX3icdVJdb9MwFHW7wkaB0QEvCIEsqklFgqqZ+HpBGgMhHoug26QkixzHWa05drCdbsXyO/+K38Ejf4DfgJO2iPTDUpSjc8+991z7xjmjSg8GvxrNrda169s7N9o3b93evdPZu3usRCExGWHBhDyNkSKMcjLSVDNymkuCspiRk/jifRk/mRCpqOBf9TQnYYbOOU0pRtpRUedHGgXdIEN6HMfwKDLBBEmSK8oEt2e4Z4KqhcFTxO1CNrRP4dsg8PsHJAvhpmSDYYCpxDA9M889a3tlJ7ipYDvqdAf9QXXgKvDmoAvmZxjtNX8GicBFRrjGDCnle4NchwZJTTEjth0UiuQIX6Bz4jvIUUbUs2RCc1XB0FROLNx3wQSmQrqPa1ix/ycblCk1zWKnLP2q5VhJrov5hU7fhIbyvNCE41mjtGBQC1g+BUyoJFizqQMIS+psQzxGEmHtHqzW5RKpqbNQm8mUDbUQTNn1jtbPUKex+lYITVZLlFehVvtJla5hE7GsTaoCde4qdaPZdnsfTtzYohzxA3EvJ8kX50ywjy7DxA4k1ozsAmXWcLtG+Y7lYxQTbeZLVIlnv3bAySUWWYZ4YgJFs5yRK+t7oXFlmEaR6bp9rKtKSzPJv3IbVEIK7i7Maf1wxhjPbiop5HciRV09WKjdynvLC74Kjg/63qv+y88vuodH8+XfAQ/BE9ADHngNDsEnMAQjgMGfxv3Go8bjrd+t7dZuqzOTNhvznHugdloP/gLl1n+Z</latexit> f# Bc " (P) = Bc f 1 " (f# P) <latexit sha1_base64="oxwIwTMBhjjfrVxoFxp/ld0q1TY=">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</latexit> P <latexit sha1_base64="grLMZw/pMTk4T1azH7U2vgJPe6U=">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</latexit> f# P <latexit sha1_base64="RujgIW5hbCfPt+p+9te8vTBouXo=">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</latexit> Bc " (P)
  9. This Propagation Result Enables ... <latexit sha1_base64="Wq8NwA30p7/Dsv6SP+WfBDe+37c=">AAAB+3icbVBNSwMxEM3Wr1q/aj16CRbBU9ktRT0WPOixgv2Adimz2WwbmmSXJCuWpX/FiwdFvPpHvPlvTNs9aOuDgcd7M8zMCxLOtHHdb6ewsbm1vVPcLe3tHxwelY8rHR2nitA2iXmsegFoypmkbcMMp71EURABp91gcjP3u49UaRbLBzNNqC9gJFnECBgrDcuVgY4iEIxP8S2kWjOwYtWtuQvgdeLlpIpytIblr0EYk1RQaQgHrfuemxg/A2UY4XRWGqSaJkAmMKJ9SyUIqv1scfsMn1slxFGsbEmDF+rviQyE1lMR2E4BZqxXvbn4n9dPTXTtZ0wmqaGSLBdFKccmxvMgcMgUJcb+HTIgitlbMRmDAmJsXCUbgrf68jrp1GveZa1xX682G3kcRXSKztAF8tAVaqI71EJtRNATekav6M2ZOS/Ou/OxbC04+cwJ+gPn8wcOiJRq</latexit> Gaussian <latexit sha1_base64="ILk6Z3NK7/I/eUrqChJ40x5DlGI=">AAACAHicbVDLSsNAFL2pr1pfURcu3AwWwVVJpKjLghuXFewD2lIm00k7dDIJMxMhhGz8FTcuFHHrZ7jzb5ykWWjrgYHDOfc1x4s4U9pxvq3K2vrG5lZ1u7azu7d/YB8edVUYS0I7JOSh7HtYUc4E7WimOe1HkuLA47TnzW9zv/dIpWKheNBJREcBngrmM4K1kcb2SToshqQkwSIbBljPPA+1s7FddxpOAbRK3JLUoUR7bH8NJyGJAyo04VipgetEepRiqRnhNKsNY0UjTOZ4SgeGChxQNUqL3Rk6N8oE+aE0T2hUqL87UhwolQSeqcwvVMteLv7nDWLt34xSJqJYU0EWi/yYIx2iPA00YZISzRNDMJHM3IrIDEtMtMmsZkJwl7+8SrqXDfeq0bxv1lvNMo4qnMIZXIAL19CCO2hDBwhk8Ayv8GY9WS/Wu/WxKK1YZc8x/IH1+QNd1Zbc</latexit>

    P <latexit sha1_base64="64oVxapDKj95dh6Itzq7v0mOvEw=">AAACAHicbVBLSwMxGMz6rPW16sGDl2AreCq7pajHghePFewD2qVks9k2NI8lyQpl6cW/4sWDIl79Gd78N2bbPWjrQGCY+b4kM2HCqDae9+2srW9sbm2Xdsq7e/sHh+7RcUfLVGHSxpJJ1QuRJowK0jbUMNJLFEE8ZKQbTm5zv/tIlKZSPJhpQgKORoLGFCNjpaF7OtBxjDhlU1iNq1BIkd+E1NCteDVvDrhK/IJUQIHW0P0aRBKnnAiDGdK673uJCTKkDMWMzMqDVJME4Qkakb6lAnGig2weYAYvrBLBWCp7hIFz9fdGhrjWUx7aSY7MWC97ufif109NfBNkVCSpIQIvHopTBo2EeRswoopgY8NHFGFF7V8hHiOFsLGdlW0J/nLkVdKp1/yrWuO+Xmk2ijpK4Aycg0vgg2vQBHegBdoAgxl4Bq/gzXlyXpx352MxuuYUOyfgD5zPH/OOlfU=</latexit> f nonlinear <latexit sha1_base64="dQLOByCvMscQuI6eqvYWIDnIKww=">AAACInicbVDLSgNBEJz1bXxFPXoZDIKnsCviAy8BLx6NGBWyIfROenVwdnaZ6RXDkm/x4q948aCoJ8GPcfJAfBUMFFVd9HRFmZKWfP/dGxufmJyanpktzc0vLC6Vl1fObJobgQ2RqtRcRGBRSY0NkqTwIjMISaTwPLo+7PvnN2isTPUpdTNsJXCpZSwFkJPa5f3QxjEkUnV5UA0PeEh4S1FcnKRRbok3tEBDIDV1eT0HTV/RXrtc8av+APwvCUakwkY4bpdfw04q8gQ1CQXWNgM/o1YBhqRQ2CuFucUMxDVcYtNRDQnaVjE4scc3nNLhcWrc08QH6vdEAYm13SRykwnQlf3t9cX/vGZO8V6rkDrLCbUYLopzxSnl/b54RxoU5OrpSBBGur9ycQUGBLlWS66E4PfJf8nZVjXYqW7Xtyq17VEdM2yNrbNNFrBdVmNH7Jg1mGB37IE9sWfv3nv0Xry34eiYN8qssh/wPj4BRoKkwA==</latexit> 1. Robust Uncertainty Quantification <latexit sha1_base64="ofv4g/TGME3tmT2ZVE8f7HOdahw=">AAACCnicbVDNS8MwHE3n15xfVY9eokXwIKOVoV7EgRePE9wHrGWkWbqFpWlJUmGUnr34r3jxoIhX/wJv/jemXQ+6+SDweO/3S16eHzMqlW1/G5Wl5ZXVtep6bWNza3vH3N3ryCgRmLRxxCLR85EkjHLSVlQx0osFQaHPSNef3OR+94EISSN+r6Yx8UI04jSgGCktDczDYOBaMHWLm1I8RTxzQ6TGvg9bGbxyT68HpmXX7QJwkTglsUCJ1sD8cocRTkLCFWZIyr5jx8pLkVAUM5LV3ESSGOEJGpG+phyFRHppESCDx1oZwiAS+nAFC/X3RopCKaehryfzmHLey8X/vH6igksvpTxOFOF49lCQMKgimPcCh1QQrNhUE4QF1VkhHiOBsNLt1XQJzvyXF0nnrO6c1xt3DavZKOuoggNwBE6AAy5AE9yCFmgDDB7BM3gFb8aT8WK8Gx+z0YpR7uyDPzA+fwADqpnI</latexit> f# P = ? <latexit sha1_base64="ILk6Z3NK7/I/eUrqChJ40x5DlGI=">AAACAHicbVDLSsNAFL2pr1pfURcu3AwWwVVJpKjLghuXFewD2lIm00k7dDIJMxMhhGz8FTcuFHHrZ7jzb5ykWWjrgYHDOfc1x4s4U9pxvq3K2vrG5lZ1u7azu7d/YB8edVUYS0I7JOSh7HtYUc4E7WimOe1HkuLA47TnzW9zv/dIpWKheNBJREcBngrmM4K1kcb2SToshqQkwSIbBljPPA+1s7FddxpOAbRK3JLUoUR7bH8NJyGJAyo04VipgetEepRiqRnhNKsNY0UjTOZ4SgeGChxQNUqL3Rk6N8oE+aE0T2hUqL87UhwolQSeqcwvVMteLv7nDWLt34xSJqJYU0EWi/yYIx2iPA00YZISzRNDMJHM3IrIDEtMtMmsZkJwl7+8SrqXDfeq0bxv1lvNMo4qnMIZXIAL19CCO2hDBwhk8Ayv8GY9WS/Wu/WxKK1YZc8x/IH1+QNd1Zbc</latexit> P <latexit sha1_base64="TUEj9o/dKa4f5hpLeAXVgSvG+TY=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5GWwFVyUpRV0W3LisYB/QhnIznbRDZ5IwMxFqKP6KGxeKuPU/3Pk3TtostPXAwOGce+/ce/yYM6Ud59sqrK1vbG4Vt0s7u3v7B/bhUVtFiSS0RSIeya4PinIW0pZmmtNuLCkIn9OOP7nJ/M4DlYpF4b2extQTMApZwAhoIw3sk74KAhCMT3ElqOBsDMiBXXaqzhx4lbg5KaMczYH91R9GJBE01ISDUj3XibWXgtSMcDor9RNFYyATGNGeoSEIqrx0vv0MnxtliINImhdqPFd/d6QglJoK31QK0GO17GXif14v0cG1l7IwTjQNyeKjIOFYRziLAg+ZpESby4cMiGRmV0zGIIFoE1jJhOAun7xK2rWqe1mt39XKjXoeRxGdojN0gVx0hRroFjVRCxH0iJ7RK3qznqwX6936WJQWrLznGP2B9fkDd0eUjA==</latexit> f linear <latexit sha1_base64="ofv4g/TGME3tmT2ZVE8f7HOdahw=">AAACCnicbVDNS8MwHE3n15xfVY9eokXwIKOVoV7EgRePE9wHrGWkWbqFpWlJUmGUnr34r3jxoIhX/wJv/jemXQ+6+SDweO/3S16eHzMqlW1/G5Wl5ZXVtep6bWNza3vH3N3ryCgRmLRxxCLR85EkjHLSVlQx0osFQaHPSNef3OR+94EISSN+r6Yx8UI04jSgGCktDczDYOBaMHWLm1I8RTxzQ6TGvg9bGbxyT68HpmXX7QJwkTglsUCJ1sD8cocRTkLCFWZIyr5jx8pLkVAUM5LV3ESSGOEJGpG+phyFRHppESCDx1oZwiAS+nAFC/X3RopCKaehryfzmHLey8X/vH6igksvpTxOFOF49lCQMKgimPcCh1QQrNhUE4QF1VkhHiOBsNLt1XQJzvyXF0nnrO6c1xt3DavZKOuoggNwBE6AAy5AE9yCFmgDDB7BM3gFb8aT8WK8Gx+z0YpR7uyDPzA+fwADqpnI</latexit> f# P = ? <latexit sha1_base64="fUZiRlDrWa6RV2r5n5WWt7efkck=">AAAB+3icbVBNS8NAEN3Ur1q/Yj16WSyCp5KUoh4LXjxWsK3QhjLZbNqlu5uwuxFL6F/x4kERr/4Rb/4bt20O2vpg4PHeDDPzwpQzbTzv2yltbG5t75R3K3v7B4dH7nG1q5NMEdohCU/UQwiaciZpxzDD6UOqKIiQ0144uZn7vUeqNEvkvZmmNBAwkixmBIyVhm51oOMYBONTrEHYjXI0dGte3VsArxO/IDVUoD10vwZRQjJBpSEctO77XmqCHJRhhNNZZZBpmgKZwIj2LZUgqA7yxe0zfG6VCMeJsiUNXqi/J3IQWk9FaDsFmLFe9ebif14/M/F1kDOZZoZKslwUZxybBM+DwBFTlBj7d8SAKGZvxWQMCoixcVVsCP7qy+uk26j7l/XmXaPWahZxlNEpOkMXyEdXqIVuURt1EEFP6Bm9ojdn5rw4787HsrXkFDMn6A+czx9AAJSK</latexit> sampling <latexit sha1_base64="sjlPuQ12qw6GUmCDlxYS0CfJcIU=">AAAB/HicbVDLSsNAFL3xWesr2qWbwSK4KokUdVlw47KCfUAbwmQ6bYdOJmFmIsQQf8WNC0Xc+iHu/BunaRbaeuDC4Zx75849QcyZ0o7zba2tb2xubVd2qrt7+weH9tFxV0WJJLRDIh7JfoAV5UzQjmaa034sKQ4DTnvB7Gbu9x6oVCwS9zqNqRfiiWBjRrA2km/XsmxYvJKRFIv80Xfz3LfrTsMpgFaJW5I6lGj79tdwFJEkpEITjpUauE6svQxLzQineXWYKBpjMsMTOjBU4JAqLyvW5ujMKCM0jqQpoVGh/p7IcKhUGgamM8R6qpa9ufifN0j0+NrLmIgTTQVZLBonHOkIzZNAIyYp0Tw1BBPJzF8RmWKJiTZ5VU0I7vLJq6R70XAvG827Zr3VLOOowAmcwjm4cAUtuIU2dIBACs/wCm/Wk/VivVsfi9Y1q5ypwR9Ynz/PZJV8</latexit> z1 <latexit sha1_base64="r1rPK4D6ooSH9Lkti/dEyALfk9o=">AAAB/HicbVDLSsNAFL2pr1pf0S7dBIvgqiSlqMuCG5cV7APaECbTSTt0MgkzEyGG+CtuXCji1g9x5984TbPQ1gMXDufcO3fu8WNGpbLtb6Oysbm1vVPdre3tHxwemccnfRklApMejlgkhj6ShFFOeooqRoaxICj0GRn485uFP3ggQtKI36s0Jm6IppwGFCOlJc+sZ9m4eCXDKeL5o9fKc89s2E27gLVOnJI0oETXM7/GkwgnIeEKMyTlyLFj5WZIKIoZyWvjRJIY4TmakpGmHIVEulmxNrfOtTKxgkjo4soq1N8TGQqlTENfd4ZIzeSqtxD/80aJCq7djPI4UYTj5aIgYZaKrEUS1oQKghVLNUFYUP1XC8+QQFjpvGo6BGf15HXSbzWdy2b7rt3otMs4qnAKZ3ABDlxBB26hCz3AkMIzvMKb8WS8GO/Gx7K1YpQzdfgD4/MH0OqVfQ==</latexit> z2 <latexit sha1_base64="Xpi8enVimMC1uI6NqNsyZodaZlQ=">AAAB/HicbVDLSsNAFL3xWesr2qWbwSK4KokUdVlw47KCfUAbwmQ6bYdOJmFmIsQQf8WNC0Xc+iHu/BunaRbaeuDC4Zx75849QcyZ0o7zba2tb2xubVd2qrt7+weH9tFxV0WJJLRDIh7JfoAV5UzQjmaa034sKQ4DTnvB7Gbu9x6oVCwS9zqNqRfiiWBjRrA2km/XsmxYvJKRFIv80Rd57tt1p+EUQKvELUkdSrR9+2s4ikgSUqEJx0oNXCfWXoalZoTTvDpMFI0xmeEJHRgqcEiVlxVrc3RmlBEaR9KU0KhQf09kOFQqDQPTGWI9VcveXPzPGyR6fO1lTMSJpoIsFo0TjnSE5kmgEZOUaJ4agolk5q+ITLHERJu8qiYEd/nkVdK9aLiXjeZds95qlnFU4ARO4RxcuIIW3EIbOkAghWd4hTfryXqx3q2PReuaVc7U4A+szx8sYZW5</latexit> zn <latexit sha1_base64="NfoFS7DONpjJanJg0jaIkAri2R4=">AAAB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JIUZcFNy4r2Ac0oUwmk3boZCbMTIQQs/BX3LhQxK2/4c6/cZpmoa0HLhzOuXfu3BMkjCrtON/Wyura+sZmbau+vbO7t28fHPaUSCUmXSyYkIMAKcIoJ11NNSODRBIUB4z0g+nNzO8/EKmo4Pc6S4gfozGnEcVIG2lkH+e5V76S4wzxwmOh0KooRnbDaTol4DJxK9IAFToj+8sLBU5jwjVmSKmh6yTaz5HUFDNS1L1UkQThKRqToaEcxUT5ebm5gGdGCWEkpCmuYan+nshRrFQWB6YzRnqiFr2Z+J83THV07eeUJ6kmHM8XRSmDWsBZGDCkkmDNMkMQltT8FeIJkghrE1ndhOAunrxMehdN97LZums12q0qjho4AafgHLjgCrTBLeiALsDgETyDV/BmPVkv1rv1MW9dsaqZI/AH1ucPkr+XEg==</latexit> . . . <latexit sha1_base64="/UcMD1bLY4/0gdda2XrivnOUZ+4=">AAAB/HicbVBNS8NAEN3Ur1q/oj16WSyCp5KUoh4LXjxWsK3QhjLZbNqlm82yuxFCqX/FiwdFvPpDvPlv3LY5aOuDgcd7M8zMCyVn2njet1Pa2Nza3invVvb2Dw6P3OOTrk4zRWiHpDxVDyFoypmgHcMMpw9SUUhCTnvh5Gbu9x6p0iwV9yaXNEhgJFjMCBgrDd3qQMcxJIznWKpUwggMHbo1r+4tgNeJX5AaKtAeul+DKCVZQoUhHLTu+540wRSUYYTTWWWQaSqBTGBE+5YKSKgOpovjZ/jcKhGOU2VLGLxQf09MIdE6T0LbmYAZ61VvLv7n9TMTXwdTJmRmqCDLRXHGsUnxPAkcMUWJsY9HDIhi9lZMxqCAGJtXxYbgr768TrqNun9Zb941aq1mEUcZnaIzdIF8dIVa6Ba1UQcRlKNn9IrenCfnxXl3PpatJaeYqaI/cD5/AAoylPw=</latexit> propagate <latexit sha1_base64="f/RjB7jlXL64Z3/Fi329RxzkPgY=">AAAB+nicbVBNSwMxEJ2tX7V+bfXoJVgET2W3FPVY8OKxgv2AdinZbLYNTbJLklVK7U/x4kERr/4Sb/4b03YP2vpg4PHeDDPzwpQzbTzv2ylsbG5t7xR3S3v7B4dHbvm4rZNMEdoiCU9UN8SaciZpyzDDaTdVFIuQ0044vpn7nQeqNEvkvZmkNBB4KFnMCDZWGrjlvo5jLBifII1FyqkeuBWv6i2A1omfkwrkaA7cr36UkExQaQjHWvd8LzXBFCvDCKezUj/TNMVkjIe0Z6nEgupgujh9hs6tEqE4UbakQQv198QUC60nIrSdApuRXvXm4n9eLzPxdTBlMs0MlWS5KM44Mgma54Aipigx9u2IYaKYvRWREVaYGJtWyYbgr768Ttq1qn9Zrd/VKo16HkcRTuEMLsCHK2jALTShBQQe4Rle4c15cl6cd+dj2Vpw8pkT+APn8wd5Z5Qa</latexit> samples <latexit sha1_base64="NfoFS7DONpjJanJg0jaIkAri2R4=">AAAB/3icbVDLSsNAFJ34rPUVFdy4GSyCq5JIUZcFNy4r2Ac0oUwmk3boZCbMTIQQs/BX3LhQxK2/4c6/cZpmoa0HLhzOuXfu3BMkjCrtON/Wyura+sZmbau+vbO7t28fHPaUSCUmXSyYkIMAKcIoJ11NNSODRBIUB4z0g+nNzO8/EKmo4Pc6S4gfozGnEcVIG2lkH+e5V76S4wzxwmOh0KooRnbDaTol4DJxK9IAFToj+8sLBU5jwjVmSKmh6yTaz5HUFDNS1L1UkQThKRqToaEcxUT5ebm5gGdGCWEkpCmuYan+nshRrFQWB6YzRnqiFr2Z+J83THV07eeUJ6kmHM8XRSmDWsBZGDCkkmDNMkMQltT8FeIJkghrE1ndhOAunrxMehdN97LZums12q0qjho4AafgHLjgCrTBLeiALsDgETyDV/BmPVkv1rv1MW9dsaqZI/AH1ucPkr+XEg==</latexit> . . . <latexit sha1_base64="NXtSR7ZStVrXUmwwGIob+IUxyBg=">AAAB/XicbVDLSsNAFL2pr1pf8bFzEyxC3ZREirosuHFZwT6gDWEynbRDJ5MwMxHSUPwVNy4Ucet/uPNvnKZZaOuBC4dz7p079/gxo1LZ9rdRWlvf2Nwqb1d2dvf2D8zDo46MEoFJG0csEj0fScIoJ21FFSO9WBAU+ox0/cnt3O8+EiFpxB9UGhM3RCNOA4qR0pJnngS1bJA/k+EU8dnUc2YXnlm163YOa5U4BalCgZZnfg2GEU5CwhVmSMq+Y8fKzZBQFDMyqwwSSWKEJ2hE+ppyFBLpZvnamXWulaEVREIXV1au/p7IUChlGvq6M0RqLJe9ufif109UcONmlMeJIhwvFgUJs1RkzaOwhlQQrFiqCcKC6r9aeIwEwkoHVtEhOMsnr5LOZd25qjfuG9Vmo4ijDKdwBjVw4BqacActaAOGKTzDK7wZT8aL8W58LFpLRjFzDH9gfP4Ak8OVRQ==</latexit> f(z1) <latexit sha1_base64="ZsQtWDSZk+wcG0V001QrySa7U3w=">AAAB/XicbVDLSsNAFL3xWesrPnZugkWom5KUoi4LblxWsA9oQ5hMJ+3QySTMTIQ0FH/FjQtF3Pof7vwbp2kW2nrgwuGce+fOPX7MqFS2/W2srW9sbm2Xdsq7e/sHh+bRcUdGicCkjSMWiZ6PJGGUk7aiipFeLAgKfUa6/uR27ncfiZA04g8qjYkbohGnAcVIackzT4NqNsifyXCK+Gzq1WeXnlmxa3YOa5U4BalAgZZnfg2GEU5CwhVmSMq+Y8fKzZBQFDMyKw8SSWKEJ2hE+ppyFBLpZvnamXWhlaEVREIXV1au/p7IUChlGvq6M0RqLJe9ufif109UcONmlMeJIhwvFgUJs1RkzaOwhlQQrFiqCcKC6r9aeIwEwkoHVtYhOMsnr5JOveZc1Rr3jUqzUcRRgjM4hyo4cA1NuIMWtAHDFJ7hFd6MJ+PFeDc+Fq1rRjFzAn9gfP4AlUmVRg==</latexit> f(z2) <latexit sha1_base64="PfWkEnZr2CcTZtR76NMq5k45ysA=">AAAB/XicbVDLSsNAFL2pr1pf8bFzEyxC3ZREirosuHFZwT6gDWEynbRDJ5MwMxHSUPwVNy4Ucet/uPNvnKZZaOuBC4dz7p079/gxo1LZ9rdRWlvf2Nwqb1d2dvf2D8zDo46MEoFJG0csEj0fScIoJ21FFSO9WBAU+ox0/cnt3O8+EiFpxB9UGhM3RCNOA4qR0pJnngS1bJA/k+EU8dnU47MLz6zadTuHtUqcglShQMszvwbDCCch4QozJGXfsWPlZkgoihmZVQaJJDHCEzQifU05Col0s3ztzDrXytAKIqGLKytXf09kKJQyDX3dGSI1lsveXPzP6ycquHEzyuNEEY4Xi4KEWSqy5lFYQyoIVizVBGFB9V8tPEYCYaUDq+gQnOWTV0nnsu5c1Rv3jWqzUcRRhlM4gxo4cA1NuIMWtAHDFJ7hFd6MJ+PFeDc+Fq0lo5g5hj8wPn8A8LGVgg==</latexit> f(zn) <latexit sha1_base64="0yahE/JVHvGXRHlKvxmT9u4OUWc=">AAAB/3icbVDLSsNAFJ3UV62vqODGzWARXJWkFHVZcOOygn1AG8pkMmmHziPMTIQQu/BX3LhQxK2/4c6/cdpmoa0HLhzOuZd77wkTRrXxvG+ntLa+sblV3q7s7O7tH7iHRx0tU4VJG0smVS9EmjAqSNtQw0gvUQTxkJFuOLmZ+d0HojSV4t5kCQk4GgkaU4yMlYbuyUDHMeKUZVDJMNWGxhkVo6Fb9WreHHCV+AWpggKtofs1iCROOREGM6R13/cSE+RIGYoZmVYGqSYJwhM0In1LBeJEB/n8/ik8t0oEY6lsCQPn6u+JHHGtMx7aTo7MWC97M/E/r5+a+DrIqUhSQwReLIpTBo2EszBgRBXBxv4eUYQVtbdCPEYKYWMjq9gQ/OWXV0mnXvMva427erXZKOIog1NwBi6AD65AE9yCFmgDDB7BM3gFb86T8+K8Ox+L1pJTzByDP3A+fwCrBJZ8</latexit> robustifying <latexit sha1_base64="LlHqRyKTpQIIfWimFHozG8+QVvE=">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</latexit> Bc " 1 n n X i=1 zi ! <latexit sha1_base64="/UcMD1bLY4/0gdda2XrivnOUZ+4=">AAAB/HicbVBNS8NAEN3Ur1q/oj16WSyCp5KUoh4LXjxWsK3QhjLZbNqlm82yuxFCqX/FiwdFvPpDvPlv3LY5aOuDgcd7M8zMCyVn2njet1Pa2Nza3invVvb2Dw6P3OOTrk4zRWiHpDxVDyFoypmgHcMMpw9SUUhCTnvh5Gbu9x6p0iwV9yaXNEhgJFjMCBgrDd3qQMcxJIznWKpUwggMHbo1r+4tgNeJX5AaKtAeul+DKCVZQoUhHLTu+540wRSUYYTTWWWQaSqBTGBE+5YKSKgOpovjZ/jcKhGOU2VLGLxQf09MIdE6T0LbmYAZ61VvLv7n9TMTXwdTJmRmqCDLRXHGsUnxPAkcMUWJsY9HDIhi9lZMxqCAGJtXxYbgr768TrqNun9Zb941aq1mEUcZnaIzdIF8dIVa6Ba1UQcRlKNn9IrenCfnxXl3PpatJaeYqaI/cD5/AAoylPw=</latexit> propagate <latexit sha1_base64="btrbTeUsNJxkdUBTHvyfwMfkEAM=">AAAB/XicbVDLSgNBEOyNrxhf6+PmZTAInsJuCOox4MVjBPOAZAmzs7PJkJnZZWZWiCH4K148KOLV//Dm3zhJ9qCJBQ1FVTfdXWHKmTae9+0U1tY3NreK26Wd3b39A/fwqKWTTBHaJAlPVCfEmnImadMww2knVRSLkNN2OLqZ+e0HqjRL5L0ZpzQQeCBZzAg2Vuq7Jz0dx1gwPkYqCTNtJNW675a9ijcHWiV+TsqQo9F3v3pRQjJBpSEca931vdQEE6wMI5xOS71M0xSTER7QrqUSC6qDyfz6KTq3SoTiRNmSBs3V3xMTLLQei9B2CmyGetmbif953czE18GEyTQzVJLFojjjyCRoFgWKmKLE2M8jholi9lZEhlhhYmxgJRuCv/zyKmlVK/5lpXZXLddreRxFOIUzuAAfrqAOt9CAJhB4hGd4hTfnyXlx3p2PRWvByWeO4Q+czx8ZOJWb</latexit> robustness <latexit sha1_base64="ILk6Z3NK7/I/eUrqChJ40x5DlGI=">AAACAHicbVDLSsNAFL2pr1pfURcu3AwWwVVJpKjLghuXFewD2lIm00k7dDIJMxMhhGz8FTcuFHHrZ7jzb5ykWWjrgYHDOfc1x4s4U9pxvq3K2vrG5lZ1u7azu7d/YB8edVUYS0I7JOSh7HtYUc4E7WimOe1HkuLA47TnzW9zv/dIpWKheNBJREcBngrmM4K1kcb2SToshqQkwSIbBljPPA+1s7FddxpOAbRK3JLUoUR7bH8NJyGJAyo04VipgetEepRiqRnhNKsNY0UjTOZ4SgeGChxQNUqL3Rk6N8oE+aE0T2hUqL87UhwolQSeqcwvVMteLv7nDWLt34xSJqJYU0EWi/yYIx2iPA00YZISzRNDMJHM3IrIDEtMtMmsZkJwl7+8SrqXDfeq0bxv1lvNMo4qnMIZXIAL19CCO2hDBwhk8Ayv8GY9WS/Wu/WxKK1YZc8x/IH1+QNd1Zbc</latexit> P <latexit sha1_base64="uFraGWyBysa7cS4l5xCBG6MeIP8=">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</latexit> Bc f 1 " 1 n n X i=1 f(zi) ! <latexit sha1_base64="mPIxxLu67+WdTn407OsMQN1owo4=">AAACGnicbVDLSgNBEJz1bXxFPXoZDIIghF0J6jGgBy+CijGBJITeSa8Ozs4sM71CCPkOL/6KFw+KeBMv/o2TmIOvgoGiqpqerjhT0lEYfgQTk1PTM7Nz84WFxaXlleLq2qUzuRVYE0YZ24jBoZIaayRJYSOzCGmssB7fHA79+i1aJ42+oF6G7RSutEykAPJSpxi1XJJAKlWPnxhNyA/BKsN3+JHfbWWcD2Og+LmJc0canesUS2E5HIH/JdGYlNgYp53iW6trRJ6iJqHAuWYUZtTugyUpFA4KrdxhBuIGrrDpqYYUXbs/Om3At7zS5Ymx/mniI/X7RB9S53pp7JMp0LX77Q3F/7xmTslBuy91lhNq8bUoyRUnw4c98a60KMjX0pUgrPR/5eIaLAjybRZ8CdHvk/+Sy91ytFeunO2WqpVxHXNsg22ybRaxfVZlx+yU1Zhgd+yBPbHn4D54DF6C16/oRDCeWWc/ELx/AlPPoQk=</latexit> Monte Carlo + Distributional Robustness 10 Q: connection to … 
 particle filter ?
  10. <latexit sha1_base64="bbVgCtAW6+3v6Z8vrMxoO3cnweg=">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</latexit> Bc D† k " 1 n n X

    i=1 Ak+1x0+Dk b w(i) ! Propagation: <latexit sha1_base64="02MLkzBvNby7OyXw169UXxPt2qk=">AAAEQXicdVLLbhMxFHUSHiW8UliysUgrIYGiTCUey/IQYllE01bKRJHHc6dx47EH25NmsPxtLPgBtnwCO8SCDQvwTFLEJKml0Ryde3zvudc3yjjTpt//1mi2rly9dn3rRvvmrdt37na27x1pmSsKAyq5VCcR0cCZgIFhhsNJpoCkEYfjaPq6jB/PQGkmxaEpMhil5FSwhFFiPDXunIU6SUjKeIEPJ4BjX1GxKC+DhONcUFCGMGEKLBNsvEIbYgDv2LCqbSOeg5uP7fRx4NwOZhrbMEowzAk1vHCYkszkCmIcFeNOt9/rVwevg2AJumh5Dsbbzc9hLGmegjCUE62HQT8zI0uUYZSDa4e5hozQKTmFoYeCpKCfxDOW6QqObOXR4V0fjHEilf+EwRX7/2VLUq2LNPLKlJiJXo2V5KbYMDfJi5FlIssNCLoolOQcG4nLYftxKijn4AGhinnbmE6I8rPxT1Krck504S3UerJlQSMl126zo8091GmqP+bSwHqKchR6vZ7SyQY2lqvauEpQ5+aJb82127t45tuWZYtvwL+cgg/emeRv/Q0beRA7O3AXKHVWuA3KlzybkAiMDUsHS/Hi1w4FnFOZpkTENtQszTjM3TAYWZ+GGzK2Xb+OdVVpaSH5l+4SlVRS+IF57XC0YGzgLksp1SdQsq7uX6j9ygerC74OjvZ6wbPe0/d73f1Xy+XfQg/QQ/QIBeg52kfv0AEaIIq+ol/oTwO1vrS+t360fi6kzcbyzn1UO63ffwHf7HqF</latexit> The distributional uncertainty of the state xk+1 is exactly captured by <latexit sha1_base64="b2kFvOBc1zYi+3kpF8EGyui27Mk=">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</latexit> 1 n n X i=1 b w(i) <latexit sha1_base64="np4ZQa4VWCf63LQypQ8+wpQohXU=">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</latexit> Bc " 1 n n X i=1 b w(i) ! <latexit sha1_base64="UxarZ4IbRL4P9Dx5+tiSMwEFljs=">AAAChnicdVFta9swEJbdbkuzt2z9uC+iodBCCXaXtWNQCPTLPnawtIXIDbIsJyKybKRzQxD6Kf1T/bZ/M8V1oa8H4h6eu+fudJdWUhiIon9BuLH55u27zlb3/YePnz73vnw9N2WtGR+zUpb6MqWGS6H4GARIfllpTotU8ot0cbqOX1xzbUSp/sKq4klBZ0rkglHw1LR3Y0lTxLIVVY4sRcbnFCwpKMzTHC/dld0T+87hXyeYSJ4DsY2b4JeFSzeNWsmBF2QlmINXMxf3xYkWszkkrSduasVJ7K78fP1oEDWGn4O4BX3U2tm0d0uyktUFV8AkNWYSRxUklmoQTHLXJbXhFWULOuMTDxUtuElsM57Du57JcF5q/xTghn2osLQwZlWkPnO9HvM0tiZfik1qyH8mVqiqBq7YXaO8lhhKvL4JzoTmDOTKA8q08LNiNqeaMvCX6/olxE+//BycHw7io8Hwz7A/Grbr6KBvaAftoRgdoxH6jc7QGLFgI9gPDoPvYScchD/C47vUMGg12+iRhaP/8mTHEg==</latexit> b w(i) := nh b w(i) 0 , . . . , b w(i) k ion i=1 <latexit sha1_base64="i1zMCpKWVtMiKqNNZLj50wlnW0A=">AAACCXicbVDLSgMxFM3UV62vUZdugq3gqsyUoi4LblxWsA9oh5JJ77SxmWRIMkIp3brxV9y4UMStf+DOvzFtZ6GtBwKHc+7l5pww4Uwbz/t2cmvrG5tb+e3Czu7e/oF7eNTUMlUUGlRyqdoh0cCZgIZhhkM7UUDikEMrHF3P/NYDKM2kuDPjBIKYDASLGCXGSj0Xd3UUkZjxMS6JEhaSacBGkXugRioGuucWvbI3B14lfkaKKEO95351+5KmMQhDOdG643uJCSZEGUY5TAvdVENC6IgMoGOpIDHoYDJPMsVnVunjSCr7hMFz9ffGhMRaj+PQTsbEDPWyNxP/8zqpia6CCRNJakDQxaEo5dhIPKsF95myiW0LfUaoYvavmA6JItTY8gq2BH858ippVsr+Rbl6WynWqlkdeXSCTtE58tElqqEbVEcNRNEjekav6M15cl6cd+djMZpzsp1j9AfO5w//6Jne</latexit> n noise trajectories Capture uncertainty: 11 <latexit sha1_base64="Pt6UxiggGuAQ8oUD9vTGl2Acqt4=">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</latexit> xk+1 = Axk + wk <latexit sha1_base64="hIIxfXgM/4D58y8oaFXFz6CuKyk=">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</latexit> x0 <latexit sha1_base64="FySDpZPbIISG0ZLRJLGzd+PtCFA=">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</latexit> wk <latexit sha1_base64="KES6+a8TUKTsVceUMLRdtmbM08Y=">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</latexit> xk+1 <latexit sha1_base64="7ZvfjPQnDJE2l8oncKY9wz2Gcq8=">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</latexit> xk+1 = Axk + wk = Ak+1x0 + ⇥ Ak . . . A I ⇤ | {z } Dk (full row-rank) 2 4 w0 . . . wk 3 5 <latexit sha1_base64="omwh5aOAO0gItjf/aImXdF85BgY=">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</latexit> xk+1 = Axk + wk = Ak+1x0 + ⇥ Ak . . . A I ⇤ 2 4 w0 . . . wk 3 5 This Propagation Result Enables ... 2. Distributionally Robust System Theory
  11. <latexit sha1_base64="NTs81QWMntz8PzuMxJ8eqrkLvpw=">AAACBXicbVDLSsNAFJ34rPUVdamLwSIIQkmkqMuCG5cV7APaECaTm3bo5MHMRKghGzf+ihsXirj1H9z5N07TLLT1wIXDOffOnXu8hDOpLOvbWFpeWV1br2xUN7e2d3bNvf2OjFNBoU1jHoueRyRwFkFbMcWhlwggoceh642vp373HoRkcXSnJgk4IRlGLGCUKC255tHAB66Imw2KtzKPp5A/uNn4zM5z16xZdasAXiR2SWqoRMs1vwZ+TNMQIkU5kbJvW4lyMiIUoxzy6iCVkBA6JkPoaxqREKSTFZtzfKIVHwex0BUpXKi/JzISSjkJPd0ZEjWS895U/M/rpyq4cjIWJamCiM4WBSnHKsbTSLDPBFDFJ5oQKpj+K6YjIghVOriqDsGeP3mRdM7r9kW9cduoNRtlHBV0iI7RKbLRJWqiG9RCbUTRI3pGr+jNeDJejHfjY9a6ZJQzB+gPjM8fgVqZLg==</latexit> zk+1 <latexit sha1_base64="LTO2KvHs6v7QW747eENU4GWbPCQ=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiRS1GXBjcsK9gFNCJPJbTt0MgkzE6GGgBt/xY0LRdz6E+78G6dpFtp64MLhnHvnzj1BwqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaIfYAmMcugoqhj0EwE4Chj0gsn1zO/dg5A05ndqmoAX4RGnQ0qw0pJvHrkhMIX9zC3eygKWQv7gZ5M898263bALWMvEKUkdlWj75pcbxiSNgCvCsJQDx06Ul2GhKGGQ19xUQoLJBI9goCnHEUgvK/bm1qlWQmsYC11cWYX6eyLDkZTTKNCdEVZjuejNxP+8QaqGV15GeZIq4GS+aJgyS8XWLBArpAKIYlNNMBFU/9UiYywwUTq2mg7BWTx5mXTPG85Fo3nbrLeaZRxVdIxO0Bly0CVqoRvURh1E0CN6Rq/ozXgyXox342PeWjHKmUP0B8bnD5ZYmL4=</latexit> zk <latexit sha1_base64="XVziT2hJ1zoQnfQSS8m7R6agyZc=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4sSRS1GXBjcsK9gFtCJPJTTt08mBmItSQjRt/xY0LRdz6D+78G6dpFtp64MLhnHvnzj1ewplUlvVtLC2vrK6tVzaqm1vbO7vm3n5Hxqmg0KYxj0XPIxI4i6CtmOLQSwSQ0OPQ9cbXU797D0KyOLpTkwSckAwjFjBKlJZc82jgA1fEzQbFW5nHU8gf3Gx8Zue5a9asulUALxK7JDVUouWaXwM/pmkIkaKcSNm3rUQ5GRGKUQ55dZBKSAgdkyH0NY1ICNLJis05PtGKj4NY6IoULtTfExkJpZyEnu4MiRrJeW8q/uf1UxVcORmLklRBRGeLgpRjFeNpJNhnAqjiE00IFUz/FdMREYQqHVxVh2DPn7xIOud1+6LeuG3Umo0yjgo6RMfoFNnoEjXRDWqhNqLoET2jV/RmPBkvxrvxMWtdMsqZA/QHxucPhGiZMA==</latexit> zk 1

    <latexit sha1_base64="UkycVXA30mQJf6VQNB3eE1NKQDU=">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</latexit> Bc D† k 2 " 1 n n X i=1 b e(i) k 1 ! <latexit sha1_base64="q/IhJOpIWAi3U6KAgLuJT2XnhXM=">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</latexit> Bc D† k 1 " 1 n n X i=1 b e(i) k ! <latexit sha1_base64="xLlJDwZXaBurwMn4iNP+I3bVzMs=">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</latexit> Bc D† k " 1 n n X i=1 b e(i) k+1 ! <latexit sha1_base64="xq0ID4IB6qYVdextWQznJzIrNVo=">AAACE3icbVBNSwMxEM36bf2qevQSrIIolN1S1KPgxWMFq0Jbymw6W0OT7JJkhWXpf/DiX/HiQRGvXrz5b0zbPWj1wcDjvZlM5oWJ4Mb6/pc3Mzs3v7C4tFxaWV1b3yhvbl2bONUMmywWsb4NwaDgCpuWW4G3iUaQocCbcHA+8m/uURseqyubJdiR0Fc84gysk7rlw7aJIpBcZNSATAQaGkd0rz1+OWcZqCF288FRMNzrlit+1R+D/iVBQSqkQKNb/mz3YpZKVJYJMKYV+Int5KAtZwKHpXZqMAE2gD62HFUg0XTy8eYh3XdKj0axdqUsHas/J3KQxmQydJ0S7J2Z9kbif14rtdFpJ+cqSS0qNlkUpYLamI4Coj2ukVmXR48D09z9lbI70MCsi7HkQgimT/5LrmvV4Lhav6xVzupFHEtkh+ySAxKQE3JGLkiDNAkjD+SJvJBX79F79t6890nrjFfMbJNf8D6+AdIFnhI=</latexit> samples of ek+1 12 <latexit sha1_base64="2gNVHEsPbLQhErTs9RmehnfCTys=">AAAB/nicbVDLSsNAFJ3UV62vqLhyEyyCIJREirosuHFZwT6gDWEyvW2HTiZhZiLUIeCvuHGhiFu/w51/4zTNQlsPXDicc+/cuSdMGJXKdb+t0srq2vpGebOytb2zu2fvH7RlnAoCLRKzWHRDLIFRDi1FFYNuIgBHIYNOOLmZ+Z0HEJLG/F5NE/AjPOJ0SAlWRgrsI93PH9EhSyF7DPTk3MuywK66NTeHs0y8glRRgWZgf/UHMUkj4IowLGXPcxPlaywUJQyySj+VkGAywSPoGcpxBNLX+ebMOTXKwBnGwhRXTq7+ntA4knIahaYzwmosF72Z+J/XS9Xw2teUJ6kCTuaLhilzVOzMsnAGVABRbGoIJoKavzpkjAUmyiRWMSF4iycvk/ZFzbus1e/q1Ua9iKOMjtEJOkMeukINdIuaqIUI0ugZvaI368l6sd6tj3lrySpmDtEfWJ8/B76WIw==</latexit> zk+1 <latexit sha1_base64="SKxR0s2QFbc06BjBN2/w5g64Wy8=">AAACG3icbVBNSwMxEM36bf2qevQSrIIilN0i6lHw4lHBqtCtZTY7a0Oz2SXJKiXs//DiX/HiQRFPggf/jWntwa8HA4/3ZpKZF+WCa+P7H97Y+MTk1PTMbGVufmFxqbq8cq6zQjFsskxk6jICjYJLbBpuBF7mCiGNBF5EvaOBf3GDSvNMnpl+ju0UriVPOAPjpE61EeokgZSLPt2w4fA9y/ogy/CWx9gFY7Hs2N5OUF7ZLb5dlhu0U635dX8I+pcEI1IjI5x0qm9hnLEiRWmYAK1bgZ+btgVlOBNYVsJCYw6sB9fYclRCirpth7uUdNMpMU0y5UoaOlS/T1hIte6nketMwXT1b28g/ue1CpMctC2XeWFQsq+PkkJQk9FBUDTmCplxucQcmOJuV8q6oIAZF2fFhRD8PvkvOW/Ug7367mmjdrg7imOGrJF1skUCsk8OyTE5IU3CyB15IE/k2bv3Hr0X7/WrdcwbzaySH/DePwGpu6G6</latexit> b e(i) k+1 <latexit sha1_base64="3RoHHu675qYqLgNkuwEbW/QbGPA=">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</latexit> ) <latexit sha1_base64="3RoHHu675qYqLgNkuwEbW/QbGPA=">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</latexit> ) Geometry: Decomposition of Dynamics: Nominal + Error <latexit sha1_base64="uK5fLAx/2d86jkqVBXUrLPTy2TY=">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</latexit> zk+1 <latexit sha1_base64="0g7ljIyA8P3SyJRFybDY9amtWGw=">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</latexit> ek+1 <latexit sha1_base64="yXAyZJr/YBS8V9KSzlbXrHvNRj8=">AAACdXicbVFNSyNBEO2Z9TN+xfUiyEJrVAQxzkhYvQiCHvaoYFTIhKGnp5I009MzdPfohmb+wf66vfk3vHi1JwbRxIKGx3tVr6qropwzpT3v2XF/zMzOzS8s1paWV1bX6us/71RWSAptmvFMPkREAWcC2pppDg+5BJJGHO6j5LLS7x9BKpaJWz3MoZuSvmA9Rom2VFj/Z4KRiaFDIkoITXLolyU+x0FK9IASjq/CBAcR9JkwkeUk+1viL0VPoVfiIMABjzOtKjQhm+So8pwWEkuCiD98w3rDa3qjwNPAH4MGGsd1WP8fxBktUhCacqJUx/dy3TVEakY5lLWgUJATmpA+dCwUJAXVNaMZSrxnmRj3Mmmf0HjEfq4wJFVqmEY2s9qFmtQq8jutU+jeWdcwkRcaBH1v1Cs41hmuToBjJoFqPrSAUMnsrJgOiCRU20PV7BL8yS9Pg7uTpv+72bppNS5a43UsoC20gw6Qj07RBfqDrlEbUfTibDrbzo7z6v5yd93991TXGddsoC/hHr8B2t7BWw==</latexit> ek+1 = Dk 2 6 6 4 w0 . . . wk 1 wk 3 7 7 5 <latexit sha1_base64="C4JNizZPAMsMgi45ryPbC5RW70c=">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</latexit> zk+1 = Azk <latexit sha1_base64="bbVgCtAW6+3v6Z8vrMxoO3cnweg=">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</latexit> Bc D† k " 1 n n X i=1 Ak+1x0+Dk b w(i) ! <latexit sha1_base64="8wt8NpVLR5dTsC+ViYNbp0s1DQc=">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</latexit> OT ambiguity set of xk+1 = zk+1 + ek+1 : nominal: error: Q: distributional stability or contraction of 
 ambiguity sets ?
  12. This Propagation Result Enables ... <latexit sha1_base64="8DXt01n/whoFzvIazw4yAviL88I=">AAACIHicbVDLSgMxFM34rPVVdekmWARXZaYUK7gp1IXLKvYBnVIyaaYNzSRDckcsQz/Fjb/ixoUiutOvMX0stPVA4HDOvdycE8SCG3DdL2dldW19YzOzld3e2d3bzx0cNoxKNGV1qoTSrYAYJrhkdeAgWCvWjESBYM1gWJ34zXumDVfyDkYx60SkL3nIKQErdXNl34QhibgY4WLBv8Q+sAcIwvTKntY8SCZTRFj3VgWJAVxVErQS424u7xbcKfAy8eYkj+aodXOffk/RJGISqCDGtD03hk5KNHAq2DjrJ4bFhA5Jn7UtlSRippNOA47xqVV6OFTaPgl4qv7eSElkzCgK7GREYGAWvYn4n9dOILzopFzGCTBJZ4fCRGBQeNIW7nHNKNj4PU6o5vavmA6IJhRsp1lbgrcYeZk0igXvvFC6KeYrpXkdGXSMTtAZ8lAZVdA1qqE6ougRPaNX9OY8OS/Ou/MxG11x5jtH6A+c7x+yBKPr</latexit> 2. Distributionally Robust

    Control <latexit sha1_base64="SV2brZnCPHD0E9kkAFl79+nn8/k=">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</latexit> xk+1 = A xk + B uk + wk = (A + BK)k+1 x0 + Ck 2 6 6 4 v0 . . . vk 1 vk 3 7 7 5 + Dk 2 6 6 4 w0 . . . wk 1 wk 3 7 7 5 <latexit sha1_base64="i1zMCpKWVtMiKqNNZLj50wlnW0A=">AAACCXicbVDLSgMxFM3UV62vUZdugq3gqsyUoi4LblxWsA9oh5JJ77SxmWRIMkIp3brxV9y4UMStf+DOvzFtZ6GtBwKHc+7l5pww4Uwbz/t2cmvrG5tb+e3Czu7e/oF7eNTUMlUUGlRyqdoh0cCZgIZhhkM7UUDikEMrHF3P/NYDKM2kuDPjBIKYDASLGCXGSj0Xd3UUkZjxMS6JEhaSacBGkXugRioGuucWvbI3B14lfkaKKEO95351+5KmMQhDOdG643uJCSZEGUY5TAvdVENC6IgMoGOpIDHoYDJPMsVnVunjSCr7hMFz9ffGhMRaj+PQTsbEDPWyNxP/8zqpia6CCRNJakDQxaEo5dhIPKsF95myiW0LfUaoYvavmA6JItTY8gq2BH858ippVsr+Rbl6WynWqlkdeXSCTtE58tElqqEbVEcNRNEjekav6M15cl6cd+djMZpzsp1j9AfO5w//6Jne</latexit> n noise trajectories <latexit sha1_base64="B46wAeItZdjYXXkh5g+KMhic/zg=">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</latexit> b w(i) [0:k] := nh b w(i) 0 , . . . , b w(i) k ion i=1 <latexit sha1_base64="E1Th03Q9Zzrf7Qhu0Z8TVqk3euw=">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</latexit> 1 n n X i=1 b w(i) [0:k] <latexit sha1_base64="J5OWlpC+M+/HSD6ySWyNys6Q1kM=">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</latexit> Bc " 1 n n X i=1 b w(i) [0:k] ! <latexit sha1_base64="3hoekb2rYgH8st/qBdRqc6PFJCg=">AAACAHicbVDLSgMxFM3UV62vURcu3ASL4KrMlKIuC924rGAf0A4lk7nThiaZIckIpXTjr7hxoYhbP8Odf2PazkJbDwQO59zLzTlhypk2nvftFDY2t7Z3irulvf2DwyP3+KStk0xRaNGEJ6obEg2cSWgZZjh0UwVEhBw64bgx9zuPoDRL5IOZpBAIMpQsZpQYKw3cs76OYyIYn+AGSU2mAMuEaRi4Za/iLYDXiZ+TMsrRHLhf/SihmQBpKCda93wvNcGUKMMoh1mpn2lICR2TIfQslUSADqaLADN8aZUIx4myTxq8UH9vTInQeiJCOymIGelVby7+5/UyE98GUybTzICky0NxxrFJ8LwNHDEF1NjwESNUMftXTEdEEWpsZyVbgr8aeZ20qxX/ulK7r5brtbyOIjpHF+gK+egG1dEdaqIWomiGntErenOenBfn3flYjhacfOcU/YHz+QO4ipZz</latexit> Capture noise <latexit sha1_base64="d1/x9BXeKVV3B47JfmQ9BZTKshA=">AAAB/nicbVBNS8NAEN34WetXVDx5WSyCp5KUoh4LXjxWsB/QhrLZbtqlu5uwOxFCKPhXvHhQxKu/w5v/xm2bg7Y+GHi8N8PMvDAR3IDnfTtr6xubW9ulnfLu3v7BoXt03DZxqilr0VjEuhsSwwRXrAUcBOsmmhEZCtYJJ7czv/PItOGxeoAsYYEkI8UjTglYaeCe9k0UEclFhlNFmQbCFWQDt+JVvTnwKvELUkEFmgP3qz+MaSqZAiqIMT3fSyDIiQZOBZuW+6lhCaETMmI9SxWRzAT5/PwpvrDKEEextqUAz9XfEzmRxmQytJ2SwNgsezPxP6+XQnQT5FwlKTBFF4uiVGCI8SwLPOSaUbCvDzmhmttbMR0TTSjYxMo2BH/55VXSrlX9q2r9vlZp1Is4SugMnaNL5KNr1EB3qIlaiKIcPaNX9OY8OS/Ou/OxaF1zipkT9AfO5w/UfpYD</latexit> uncertainty 13 <latexit sha1_base64="ygWPfeC313R/lqfoj8nEntsjvbk=">AAACEXicbZDLSgMxFIYz9VbrrerSTbAIBaHMSFGXBTeCmwr2Am0ZMpnTNjSTGZJMsQzzCm58FTcuFHHrzp1vYzotoq0HAj/ff05O8nsRZ0rb9peVW1ldW9/Ibxa2tnd294r7B00VxpJCg4Y8lG2PKOBMQEMzzaEdSSCBx6Hlja6mfmsMUrFQ3OlJBL2ADATrM0q0QW6xfIOTbnZN4vEYUnzvjlJ8+gMl+CkeG+YWS3bFzgovC2cuSmhedbf42fVDGgcgNOVEqY5jR7qXEKkZ5ZAWurGCiNARGUDHSEECUL0k25riE0N83A+lOULjjP6eSEig1CTwTGdA9FAtelP4n9eJdf+ylzARxRoEnS3qxxzrEE/jwT6TQDWfGEGoZOatmA6JJFSbEAsmBGfxy8uieVZxzivV22qpVp3HkUdH6BiVkYMuUA1dozpqIIoe0BN6Qa/Wo/VsvVnvs9acNZ85RH/K+vgGrKedhw==</latexit> Kxk + vk <latexit sha1_base64="ZNrOCfts1FKKbes5pf3Fc9E5dow=">AAACBnicbVBPS8MwHE3nvzn/VT2KEByCp9HKUPE08OJxgpuDtpQ0S7ewNC1JOhilJy9+FS8eFPHqZ/DmtzHtetDNB4HHe79f8vKChFGpLOvbqK2srq1v1DcbW9s7u3vm/kFfxqnApIdjFotBgCRhlJOeooqRQSIIigJGHoLJTeE/TImQNOb3apYQL0IjTkOKkdKSbx675R2ZIMPcjZAaB2E2zf3Msa4nXu6bTatllYDLxK5IE1To+uaXO4xxGhGuMENSOraVKC9DQlHMSN5wU0kShCdoRBxNOYqI9LIyQg5PtTKEYSz04QqW6u+NDEVSzqJATxZJ5aJXiP95TqrCKy+jPEkV4Xj+UJgyqGJYdAKHVBCs2EwThAXVWSEeI4Gw0s01dAn24peXSf+8ZV+02nftZqdd1VEHR+AEnAEbXIIOuAVd0AMYPIJn8ArejCfjxXg3PuajNaPaOQR/YHz+AELgmZY=</latexit> v[0:k] <latexit sha1_base64="3RoHHu675qYqLgNkuwEbW/QbGPA=">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</latexit> ) <latexit sha1_base64="SV2brZnCPHD0E9kkAFl79+nn8/k=">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</latexit> xk+1 = A xk + B uk + wk = (A + BK)k+1 x0 + Ck 2 6 6 4 v0 . . . vk 1 vk 3 7 7 5 + Dk 2 6 6 4 w0 . . . wk 1 wk 3 7 7 5 3. Q: optimal 
 or robust 
 control ? <latexit sha1_base64="LZS5moBnVwgHbp+wTZFmQ3RNViQ=">AAACN3icbZDLSgMxFIYz3q23qks3waJUhDIjRUUQ6mWhCFLB2kKnlkzmtIZmMkOSEcvQt3Lja7jTjQtF3PoGphdQqz8EPv5zDifn9yLOlLbtJ2tkdGx8YnJqOjUzOze/kF5culJhLCmUaMhDWfGIAs4ElDTTHCqRBBJ4HMpe66hbL9+CVCwUl7odQS0gTcEajBJtrHr63A2IvqGE4+N6C+/tY9eDJhOJZ2zJ7jo4e7B5eLZxnbQ6eB273A+1MtB3DZy6IPzv7no6Y+fsnvBfcAaQQQMV6+lH1w9pHIDQlBOlqo4d6VpCpGaUQyflxgoiQlukCVWDggSgaknv7g5eM46PG6E0T2jcc39OJCRQqh14prN7pRqudc3/atVYN3ZrCRNRrEHQ/qJGzLEOcTdE7DMJVPO2AUIlM3/F9IZIQrWJOmVCcIZP/gtXWzlnO5e/yGcK+UEcU2gFraIsctAOKqATVEQlRNE9ekav6M16sF6sd+uj3zpiDWaW0S9Zn19qdqkx</latexit> Dk := ⇥ (A + BK)k . . . (A + BK) I ⇤ <latexit sha1_base64="j8iTuArfMPC5peQUO27pyd8hmLI=">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</latexit> Ck := ⇥ (A + BK)kB . . . (A + BK)B B ⇤ full rank <latexit sha1_base64="JuNX0bWMkmbFd0AJOZ/KAA8D2YQ=">AAACCHicbVC7SgNBFJ31GeMramnhYBCslt0Q1DJgYxnBPCAJYXZ2Nhkys7PM3BXDktLGX7GxUMTWT7Dzb5xNUmjigQuHc+7lck6QCG7A876dldW19Y3NwlZxe2d3b790cNg0KtWUNagSSrcDYpjgMWsAB8HaiWZEBoK1gtF17rfumTZcxXcwTlhPkkHMI04JWKlfOumaKCKSizHuAnuAIMrqWiXK8Nx3J/1S2XO9KfAy8eekjOao90tf3VDRVLIYqCDGdHwvgV5GNHAq2KTYTQ1LCB2RAetYGhPJTC+bBpngM6uEOFLaTgx4qv6+yIg0ZiwDuykJDM2il4v/eZ0UoqtexuMkBRbT2aMoFRgUzlvBIdeMgi0h5IRqm51iOiSaULDdFW0J/mLkZdKsuP6FW72tlGvVeR0FdIxO0Tny0SWqoRtURw1E0SN6Rq/ozXlyXpx352O2uuLMb47QHzifP76DmmY=</latexit> Proposition. <latexit sha1_base64="xbV+9pO3eQmtPm6HhKr83s7h9Mg=">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</latexit> Bc D† k " 1 n n X i=1 (A+BK)k+1x0+Ckv[0:k] +Dk b w(i) [0:k] ! <latexit sha1_base64="kBz8mEPNMEaIua+bkT/5zFdj6zw=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBZBEEoiRUUQCm5cVrQPaEOYTCft0MmDmYlYQn7FjQtF3Poj7vwbJ20W2npg4HDOvdwzx4s5k8qyvo3Syura+kZ5s7K1vbO7Z+5XOzJKBKFtEvFI9DwsKWchbSumOO3FguLA47TrTW5yv/tIhWRR+KCmMXUCPAqZzwhWWnLN6iDAaux56N5NJ6d2hq6uXbNm1a0Z0DKxC1KDAi3X/BoMI5IENFSEYyn7thUrJ8VCMcJpVhkkksaYTPCI9jUNcUClk86yZ+hYK0PkR0K/UKGZ+nsjxYGU08DTk3lSuejl4n9eP1H+pZOyME4UDcn8kJ9wpCKUF4GGTFCi+FQTTATTWREZY4GJ0nVVdAn24peXSeesbp/XG3eNWrNR1FGGQziCE7DhAppwCy1oA4EneIZXeDMy48V4Nz7moyWj2DmAPzA+fwBWapNM</latexit> S k+1 := <latexit sha1_base64="02MLkzBvNby7OyXw169UXxPt2qk=">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</latexit> The distributional uncertainty of the state xk+1 is exactly captured by
  13. Example: Distributionally Robust CVaR Constraints <latexit sha1_base64="pdgg3NHyUCvfJFb1+Sh4umKNm6w=">AAAB8nicbVDLSsNAFL2pr1pfVZduBovgqiRS1GXBjcsK9gFtKJPppB06mQkzN0IJ/Qw3LhRx69e4829M2iy09cDA4Zx7mXNPEEth0XW/ndLG5tb2Tnm3srd/cHhUPT7pWJ0YxttMS216AbVcCsXbKFDyXmw4jQLJu8H0Lve7T9xYodUjzmLuR3SsRCgYxUzqDyKKE0Yl6VWG1Zpbdxcg68QrSA0KtIbVr8FIsyTiCpmk1vY9N0Y/pQYFk3xeGSSWx5RN6Zj3M6poxK2fLiLPyUWmjEioTfYUkoX6eyOlkbWzKMgm84h21cvF/7x+guGtnwoVJ8gVW34UJpKgJvn9ZCQMZyhnGaHMiCwrYRNqKMOspbwEb/XkddK5qnvX9cZDo9ZsFHWU4QzO4RI8uIEm3EML2sBAwzO8wpuDzovz7nwsR0tOsXMKf+B8/gBY/JCb</latexit> X <latexit sha1_base64="A5pZlYmjSnVjlOB4Omh9R3qJKnY=">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</latexit>

    := ⇢ x 2 Rd : max j a> j x + bj  0 <latexit sha1_base64="nMmnaNQ7kDN0DXvaTdpTDz/kBQU=">AAACJHicbVDLSgNBEJz1bXxFPXoZjIKnsCuighfBi0cFo0ISQu+kV4fMzi4zPWJY8jFe/BUvHnzgwYvf4mTNwVfBQFHVTfVUnCtpKQzfg7Hxicmp6ZnZytz8wuJSdXnl3GbOCGyITGXmMgaLSmpskCSFl7lBSGOFF3HvaOhf3KCxMtNn1M+xncKVlokUQF7qVA9aNkkglarPuz7NyNgNDVDcaYGGQGrq8yzhG0WrTCti5XBw2+kNNjrVWlgPS/C/JBqRGhvhpFN9aXUz4VLUJBRY24zCnNoFGJJC4aDSchZzED24wqanGlK07aKMHfBNr3R5khn/NPFS/b5RQGptP439ZAp0bX97Q/E/r+ko2W8XUueOUIuvoMQpThkfNuZ7MSioLAiEkf5WLq7BgCDfa8WXEP3+8l9yvl2Pdus7p9u1w51RHTNsja2zLRaxPXbIjtkJazDB7tgDe2LPwX3wGLwGb1+jY8FoZ5X9QPDxCQxEpj4=</latexit> distributional uncertainty of xk <latexit sha1_base64="tFqHXNZW2i1Om3eJQ0S+ECVfBg8=">AAACHnicbVDJSgNBFOxxjXGLevTSmAiewkyIyzHgxZsRs0FmCD2dN0mTnoXuN0II+RIv/ooXD4oInvRv7CwHTSxoKKpe0e+Vn0ih0ba/rZXVtfWNzcxWdntnd28/d3DY0HGqONR5LGPV8pkGKSKoo0AJrUQBC30JTX9wPfGbD6C0iKMaDhPwQtaLRCA4QyN1cueuDgIWCjmktzVqcqKXChxSDUgLbsiw7/v0vjMoUM4STBVoin3o5PJ20Z6CLhNnTvJkjmon9+l2Y56GECGXTOu2YyfojZhCwSWMs26qIWF8wHrQNjRiIWhvND1vTE+N0qVBrMyLkE7V34kRC7Uehr6ZnCysF72J+J/XTjG48kYiSlKEiM8+ClJJMaaTrmhXKOBoqukKxpUwu1LeZ4pxNI1mTQnO4snLpFEqOhfF8l0pXynP68iQY3JCzohDLkmF3JAqqRNOHskzeSVv1pP1Yr1bH7PRFWueOSJ/YH39ADOVod4=</latexit> OT ambiguity set S k captures the <latexit sha1_base64="8TdWYF2RuuoMfFKpl40wOUXoglE=">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</latexit> CVaRQ maxj a> j xk + bj is of the form: <latexit sha1_base64="stkysKKWiYvwSPaCKniEM3RHjUU=">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</latexit> inf ✓2⇥ E z⇠Q [`(✓, z)] <latexit sha1_base64="0aQLLm2OeAIYT3ifkUmEunkA0VI=">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</latexit> with `(✓, z) = max of a ne functions <latexit sha1_base64="4H3bNdlB/cm8GVulqUMb7U42Eo0=">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</latexit> <latexit sha1_base64="/c3QrfpBW7593Ira4NUVgOPws/U=">AAACB3icbVBNS8NAEN34WetX1KMgi0XwVJJSVDwVevFYwX5AG8pms2mXbjZhdyKW0JsX/4oXD4p49S9489+4bXPQ1gcDj/dmmJnnJ4JrcJxva2V1bX1js7BV3N7Z3du3Dw5bOk4VZU0ai1h1fKKZ4JI1gYNgnUQxEvmCtf1Rfeq375nSPJZ3ME6YF5GB5CGnBIzUt096OgxJxMUY94A9gB9m9VhqUIRLmFz37ZJTdmbAy8TNSQnlaPTtr14Q0zRiEqggWnddJwEvIwo4FWxS7KWaJYSOyIB1DZUkYtrLZn9M8JlRAhzGypQEPFN/T2Qk0noc+aYzIjDUi95U/M/rphBeeRmXSQpM0vmiMBUYYjwNBQdcMQomg4ATqri5FdMhUYSCia5oQnAXX14mrUrZvShXbyulWjWPo4CO0Sk6Ry66RDV0gxqoiSh6RM/oFb1ZT9aL9W59zFtXrHzmCP2B9fkD2yeZ5w==</latexit> Constraint: <latexit sha1_base64="7b7W7y8OeJYhQxyUDZpPhiKsw7g=">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</latexit> sup Q2S k CVaRQ ✓ max j a> j xk + bj ◆  0 <latexit sha1_base64="4H3bNdlB/cm8GVulqUMb7U42Eo0=">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</latexit> <latexit sha1_base64="7qHQVTC5D9BdtoK1l3/8cyCfxkg=">AAAB73icbVBNS8NAEJ3Ur1q/qh69LBbBiyWRoh4LXjxWsB/QhrLZbNqlm03cnQil9E948aCIV/+ON/+N2zYHbX0w8Hhvhpl5QSqFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZZJMM95kiUx0J6CGS6F4EwVK3kk1p3EgeTsY3c789hPXRiTqAccp92M6UCISjKKVOt5FL+QSab9ccavuHGSVeDmpQI5Gv/zVCxOWxVwhk9SYruem6E+oRsEkn5Z6meEpZSM64F1LFY258Sfze6fkzCohiRJtSyGZq78nJjQ2ZhwHtjOmODTL3kz8z+tmGN34E6HSDLlii0VRJgkmZPY8CYXmDOXYEsq0sLcSNqSaMrQRlWwI3vLLq6R1WfWuqrX7WqVey+Mowgmcwjl4cA11uIMGNIGBhGd4hTfn0Xlx3p2PRWvByWeO4Q+czx9rDo+J</latexit> 1 <latexit sha1_base64="RfWWe8dJok1B+tsUylLZo3xBHq4=">AAAB9XicbVBNSwMxEJ31s9avqkcvwSJ4KrulqMdCLx6r2A9o15LNZtvQJLskWaUs/R9ePCji1f/izX9j2u5BWx8MPN6bYWZekHCmjet+O2vrG5tb24Wd4u7e/sFh6ei4reNUEdoiMY9VN8CaciZpyzDDaTdRFIuA004wbsz8ziNVmsXy3kwS6gs8lCxiBBsrPfR1FGHB+AQ12vhuUCq7FXcOtEq8nJQhR3NQ+uqHMUkFlYZwrHXPcxPjZ1gZRjidFvuppgkmYzykPUslFlT72fzqKTq3SoiiWNmSBs3V3xMZFlpPRGA7BTYjvezNxP+8Xmqiaz9jMkkNlWSxKEo5MjGaRYBCpigx9ueQYaKYvRWREVaYGBtU0YbgLb+8StrVindZqd1Wy/VaHkcBTuEMLsCDK6jDDTShBQQUPMMrvDlPzovz7nwsWtecfOYE/sD5/AH7jZIi</latexit> CVaR <latexit sha1_base64="Sx7nk9Mr803qftLk/29oeQCCbZA=">AAACFXicbZDJSgNBEIZ7XGPcoh69NCaCgoaZENRjwIvHCEYDSQg1PTWmsWehu0YMIS/hxVfx4kERr4I338bOcnAraPj5/yqq6/NTJQ257qczMzs3v7CYW8ovr6yurRc2Ni9NkmmBDZGoRDd9MKhkjA2SpLCZaoTIV3jl35yO8qtb1EYm8QX1U+xEcB3LUAoga3ULB20ThhBJ1ed7eJeioHHAk5BTD3mpHaAiKB0SSLXfLRTdsjsu/ld4U1Fk06p3Cx/tIBFZhDEJBca0PDelzgA0SaFwmG9nBlMQN3CNLStjiNB0BuOrhnzXOgEPE21fTHzsfp8YQGRMP/JtZwTUM7+zkflf1sooPOkMZJxmhLGYLAozxSnhI0Q8kNpysEQCCUJL+1cueqBBkAWZtxC83yf/FZeVsndUrp5XirXqFEeObbMdtsc8dsxq7IzVWYMJds8e2TN7cR6cJ+fVeZu0zjjTmS32o5z3Ly70niM=</latexit> (expectation of the -tail) <latexit sha1_base64="4H3bNdlB/cm8GVulqUMb7U42Eo0=">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</latexit> <latexit sha1_base64="oeGIjxy1z9e68fF/gMLwuDjXqs0=">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</latexit> Pr (xk 2 X) 1 & benign tail risk 15
  14. Distributionally Robust Optimization (DRO) <latexit sha1_base64="kPWmKAe8K9cV7HHZu+W2wpiw7iA=">AAACE3icbVDLSgMxFM3UV62vqks3wSKIizJTirosuHFZ0T6gLSWTybShSWZI7ohl6D+48VfcuFDErRt3/o2ZtgttPRA4nHMuN/f4seAGXPfbya2srq1v5DcLW9s7u3vF/YOmiRJNWYNGItJtnxgmuGIN4CBYO9aMSF+wlj+6yvzWPdOGR+oOxjHrSTJQPOSUgJX6xbOuCUMiuRjjLrAH8MP0FnSkBjhIiMBhpGUiptlJv1hyy+4UeJl4c1JCc9T7xa9uENFEMgVUEGM6nhtDLyUaOBVsUugmhsWEjsiAdSxVRDLTS6c3TfCJVYJsv30K8FT9PZESacxY+jYpCQzNopeJ/3mdBMLLXspVnABTdLYoTASGCGcF4YBrRsH2EXBCNbd/xXRINKFgayzYErzFk5dJs1L2zsvVm0qpVp3XkUdH6BidIg9doBq6RnXUQBQ9omf0it6cJ+fFeXc+ZtGcM585RH/gfP4AYm2fEw==</latexit> Strong dual formulation <latexit

    sha1_base64="EdKdOfdX/GKSLmgEYttkobxUfUs=">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</latexit> inf 0 " + 1 n n X i=1 sup ⇣2Rd `(✓, ⇣) c(⇣ zi) <latexit sha1_base64="wBLe3qMsS+4Wbtk/w2m/G+WIa6M=">AAACAXicbVDLSgMxFM34rPU16kZwEyyCG8tMKeqy4MZlBfuAdiiZzJ02NJMZkkyxDHXjr7hxoYhb/8Kdf2PazkJbD4QczrmX5Bw/4Uxpx/m2VlbX1jc2C1vF7Z3dvX374LCp4lRSaNCYx7LtEwWcCWhopjm0Ewkk8jm0/OHN1G+NQCoWi3s9TsCLSF+wkFGijdSzj7sqDEnE+BjTWIzg4cJclIygZ5ecsjMDXiZuTkooR71nf3WDmKYRCE05UarjOon2MiI1oxwmxW6qICF0SPrQMVSQCJSXzRJM8JlRAhzG0hyh8Uz9vZGRSKlx5JvJiOiBWvSm4n9eJ9XhtZcxkaQaBJ0/FKYc6xhP68ABk0C1SR8wQiUzf8V0QCSh2pRWNCW4i5GXSbNSdi/L1btKqVbN6yigE3SKzpGLrlAN3aI6aiCKHtEzekVv1pP1Yr1bH/PRFSvfOUJ/YH3+AL4xlwo=</latexit> convex-concave <latexit sha1_base64="egOVeh9V/6PXfWhkAQEQ6KiUsjc=">AAACB3icbVBPS8MwHP11/pvzX9WjIMEheHG0Y6jHgRePE9wcbGWkabqFpWlJUmGU3bz4Vbx4UMSrX8Gb38Z060E3HwQe7/1ekt/zE86Udpxvq7Syura+Ud6sbG3v7O7Z+wcdFaeS0DaJeSy7PlaUM0HbmmlOu4mkOPI5vffH17l//0ClYrG405OEehEeChYygrWRBvZxX4UhjhifICZCJpim5wGLqMgTmA/sqlNzZkDLxC1IFQq0BvZXP4hJai7QhGOleq6TaC/DUjPC6bTSTxVNMBnjIe0ZKnBElZfN9piiU6MEKIylOUKjmfo7keFIqUnkm8kI65Fa9HLxP6+X6vDKy5hIUk0FmT8UphzpGOWloIBJSrTpIGCYSGb+isgIS0y0qa5iSnAXV14mnXrNvag1buvVZqOoowxHcAJn4MIlNOEGWtAGAo/wDK/wZj1ZL9a79TEfLVlF5hD+wPr8AZjYmb0=</latexit> infinite-dimensional <latexit sha1_base64="cQQ7eVeD9gEokkp+4/Tx6jrHiXM=">AAACLXicdVDLSgMxFM34rPVVdekmWIQKUmakqCCCoAuXClYLnTJk0lsbm3mQ3BHr0B9y46+I4EIRt/6GmbELbfVA4HDOubnJ8WMpNNr2qzUxOTU9M1uYK84vLC4tl1ZWL3WUKA51HslINXymQYoQ6ihQQiNWwAJfwpXfO878q1tQWkThBfZjaAXsOhQdwRkaySuduCBlJXXzm1IF7YGLXUA22L7foofUDdidd+Me0Czm3fwT9Eplu2rnoOPEGZIyGeLMKz277YgnAYTIJdO66dgxtlKmUHAJg6KbaIgZ77FraBoasgB0K81XD+imUdq0EylzQqS5+nMiZYHW/cA3yYBhV496mfiX10yws99KRRgnCCH/XtRJJMWIZtXRtlDAUfYNYVwJ81bKu0wxjqbgoinBGf3yOLncqTq71dp5rXxUG9ZRIOtkg1SIQ/bIETklZ6ROOHkgT+SVvFmP1ov1bn18Ryes4cwa+QXr8wvDJqkG</latexit> `(✓, z) = max j `j(✓, z) <latexit sha1_base64="IxrbsmMSaxGWOCxYuHQbUdD7Ju8=">AAACBHicbVA9SwNBEN2LXzF+RS3TLAbBKtyFoJYBG8sI+YIkhL3NXLJk9+7YnRPDkcLGv2JjoYitP8LOf+Pmo9DEBwOP92aYmefHUhh03W8ns7G5tb2T3c3t7R8cHuWPT5omSjSHBo9kpNs+MyBFCA0UKKEda2DKl9Dyxzczv3UP2ogorOMkhp5iw1AEgjO0Uj9f6JogYErICe0iPKAfpPURRBpUadrPF92SOwddJ96SFMkStX7+qzuIeKIgRC6ZMR3PjbGXMo2CS5jmuomBmPExG0LH0pApML10/sSUnltlQINI2wqRztXfEylTxkyUbzsVw5FZ9Wbif14nweC6l4owThBCvlgUJJJiRGeJ0IHQwNEGMBCMa2FvpXzENONoc8vZELzVl9dJs1zyLkuVu3KxWlnGkSUFckYuiEeuSJXckhppEE4eyTN5JW/Ok/PivDsfi9aMs5w5JX/gfP4AN/yYbA==</latexit> Theorem. <latexit sha1_base64="jfW0eVrl9r5LTIShoh5BINiwJZA=">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</latexit> inf " + 1 n Pn i=1 si s.t. ✓ 2 ⇥, 2 R +, si 2 R, ⇣ij 2 Rd, 8i, j ( `j)⇤2(✓, ⇣ij) + ⇣> ij zi + c⇤(⇣ij/ )  si, 8i, j <latexit sha1_base64="g+usfzd0ocI9alI/2svwz7/GNu4=">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</latexit> Let c be convex. Then the DRO problem has the same infimum as efficiently 
 computable 
 s.t. reasonable 
 assumptions 
 (very active area) 16 <latexit sha1_base64="S3oGoOEcUv+j5A+LXPalmgdJCXI=">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</latexit> inf ✓2⇥ sup Q2 B "(P n) E z⇠Q [`(✓, z)] <latexit sha1_base64="wuQZYDBV+n/mFTvr1xg5vcjrcD4=">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</latexit> c <latexit sha1_base64="3RoHHu675qYqLgNkuwEbW/QbGPA=">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</latexit> )
  15. Decomposition of Dynamics <latexit sha1_base64="ygWPfeC313R/lqfoj8nEntsjvbk=">AAACEXicbZDLSgMxFIYz9VbrrerSTbAIBaHMSFGXBTeCmwr2Am0ZMpnTNjSTGZJMsQzzCm58FTcuFHHrzp1vYzotoq0HAj/ff05O8nsRZ0rb9peVW1ldW9/Ibxa2tnd294r7B00VxpJCg4Y8lG2PKOBMQEMzzaEdSSCBx6Hlja6mfmsMUrFQ3OlJBL2ADATrM0q0QW6xfIOTbnZN4vEYUnzvjlJ8+gMl+CkeG+YWS3bFzgovC2cuSmhedbf42fVDGgcgNOVEqY5jR7qXEKkZ5ZAWurGCiNARGUDHSEECUL0k25riE0N83A+lOULjjP6eSEig1CTwTGdA9FAtelP4n9eJdf+ylzARxRoEnS3qxxzrEE/jwT6TQDWfGEGoZOatmA6JJFSbEAsmBGfxy8uieVZxzivV22qpVp3HkUdH6BiVkYMuUA1dozpqIIoe0BN6Qa/Wo/VsvVnvs9acNZ85RH/K+vgGrKedhw==</latexit> Kxk + vk <latexit sha1_base64="SV2brZnCPHD0E9kkAFl79+nn8/k=">AAADdnicfVJda9RAFJ1N/Kjxa1ufRJBLl2J1dUlKqb4U+uGD4EsFty1s1jCZzG6HTCZhZlK7hPwDf51v/g5ffHSSzdL90guBk3PmnnPnMmHGmdKu+6tl2Xfu3ru/8cB5+Ojxk6ftza1zleaS0D5JeSovQ6woZ4L2NdOcXmaS4iTk9CKMTyv94ppKxVLxVU8yOkzwWLARI1gbKths/fBDOmaiwJyNxZvSKfzatAh5Tku4CYq465XlIRyD/3ZZi0vowsmcIGlUQj7lZxyZYFHCd0M6h7B73D35/PpbbbrG0K0a/QTrK4I5nAYxNNOFhpPspoSFpOuqwffB51GqVYWW5CJ+Z4ZfI5gJfSqiW9+52I//iZ1d5l+5M301+HYNi8lO/ddsP2h33J5bF6wCrwEd1NRZ0P7pRynJEyo04VipgedmelhgqRnh1JjnimaYxHhMBwYKnFA1LOqJStgxTASjVJpPaKjZ+Y4CJ0pNktCcrHajlrWKXKcNcj36MCyYyHJNBZkGjXIOOoXqDULEJCWaTwzARDIzK5ArLDHR5qU6Zgne8pVXwflezzvo7X/Z6xztN+vYQC/QNtpFHnqPjtAndIb6iLR+W8+tbatj/bFf2jv2q+lRq9X0PEMLZbt/AWFEHZo=</latexit>

    xk+1 = A xk + B uk + wk = (A + BK)k+1 x0 + Ck 2 6 6 4 v0 . . . vk 1 vk 3 7 7 5 + Dk 2 6 6 4 w0 . . . wk 1 wk 3 7 7 5 <latexit sha1_base64="gZL4+CmLOMlHtE97yVPLwHW8SOM=">AAACAHicbVBNSwMxFMzWr1q/Vj148BIsghfLbinqseDFYwVbC+1S3mazbWg2WZKsUkov/hUvHhTx6s/w5r8xbfegrQOBYeY9XmbClDNtPO/bKaysrq1vFDdLW9s7u3vu/kFLy0wR2iSSS9UOQVPOBG0aZjhtp4pCEnJ6Hw6vp/79A1WaSXFnRikNEugLFjMCxko996ir4xgSxkc4zjjHSj6eKxDDnlv2Kt4MeJn4OSmjHI2e+9WNJMkSKgzhoHXH91ITjEEZRjidlLqZpimQIfRpx1IBCdXBeBZggk+tEuFYKvuEwTP198YYEq1HSWgnEzADvehNxf+8Tmbiq2DMRJoZKsj8kM2JjcTTNnDEFCXGho8YEMXsXzEZgAJibGclW4K/GHmZtKoV/6JSu62W67W8jiI6RifoDPnoEtXRDWqgJiJogp7RK3pznpwX5935mI8WnHznEP2B8/kDo5mWZQ==</latexit> full row-rank <latexit sha1_base64="ATC7kB0Sz5EmpyvoU28l+fdlICc=">AAACCXicbZDLSgMxFIYzXmu9VV26CRbBVZkpRcVVwY2Ci0qv0A4lk8m0oUlmSM4IpXTrxldx40IRt76BO9/G9LLQ1h8CP/+5hPMFieAGXPfbWVldW9/YzGxlt3d29/ZzB4cNE6easjqNRaxbATFMcMXqwEGwVqIZkYFgzWBwPak3H5g2PFY1GCbMl6SneMQpARt1c7hjoohILob4rnaLqxDTPjHAKa4ODTB51c3l3YI7FV423tzk0VyVbu6rE8Y0lUwBFcSYtucm4I+ItksFG2c7qWEJoQPSY21rFZHM+KPpJWN8apMQR7G2TwGepr8nRkQaM5SB7ZQE+maxNgn/q7VTiC79EVdJCkzR2UdRKjDEeIIFh1wzCpZCyAnVfALAgtCEgoWXtRC8xZOXTaNY8M4Lpftivlya48igY3SCzpCHLlAZ3aAKqiOKHtEzekVvzpPz4rw7H7PWFWc+c4T+yPn8AcmGmbg=</latexit> LTI Stochastic System: <latexit sha1_base64="4z74kWij8i6toGp/RrE5MZQqre0=">AAACBHicbVDLSsNAFJ3UV62vqMtuBovgqiSlqLgquHFZoS9oQ5lMJu3QmSTMQyihCzf+ihsXirj1I9z5N07SLLT1wIXDOfdy7z1+wqhUjvNtlTY2t7Z3yruVvf2DwyP7+KQnYy0w6eKYxWLgI0kYjUhXUcXIIBEEcZ+Rvj+7zfz+AxGSxlFHzRPicTSJaEgxUkYa29WRDEPEKZvDjvYJDGPBNcvNm7Fdc+pODrhO3ILUQIH22P4aBTHWnEQKMyTl0HUS5aVIKIoZWVRGWpIE4RmakKGhEeJEemn+xAKeGyXI9puKFMzV3xMp4lLOuW86OVJTuepl4n/eUKvw2ktplGhFIrxcFGoGVQyzRGBABcHKBBBQhAU1t0I8RQJhZXKrmBDc1ZfXSa9Rdy/rzftGrdUs4iiDKjgDF8AFV6AF7kAbdAEGj+AZvII368l6sd6tj2VrySpmTsEfWJ8/xVuYIw==</latexit> Tube formulation: <latexit sha1_base64="AfHKvJobHciQD5ZBrH8IPlnp7LY=">AAACInicbZDLSgMxFIYzXmu9VV26CRZBEMqMFC8LoeDGZQV7gbaUTHrahmYyQ5IRx2GexY2v4saFoq4EH8bMtIhtPRD4+f5zcpLfDThT2ra/rIXFpeWV1dxafn1jc2u7sLNbV34oKdSoz33ZdIkCzgTUNNMcmoEE4rkcGu7oKvUbdyAV88WtjgLoeGQgWJ9Rog3qFi7idnZJ7PIQkvvuKMGXeIo9pOz4l9GIiAQM6xaKdsnOCs8LZyKKaFLVbuGj3fNp6IHQlBOlWo4d6E5MpGaUQ5JvhwoCQkdkAC0jBfFAdeJsa4IPDenhvi/NERpn9O9ETDylIs81nR7RQzXrpfA/rxXq/nknZiIINQg6XtQPOdY+TvPCPSaBah4ZQahk5q2YDokkVJtU8yYEZ/bL86J+UnJOS+WbcrFSnsSRQ/voAB0hB52hCrpGVVRDFD2iZ/SK3qwn68V6tz7HrQvWZGYPTZX1/QPtT6W5</latexit> xk = zk + ek <latexit sha1_base64="9hzio5lAbKObgIrt09H1emOt4ak=">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</latexit> zk = nominal state, ek = error state <latexit sha1_base64="P4g1NbVxObGaiRXTfyTdL9+FPo0=">AAACBHicbVC7SgNBFJ31GeNr1TLNYBCswm4IKlYBLawkgnlAsoTZ2dlkyDyWmVlhCSls/BUbC0Vs/Qg7/8ZJsoUmHrhwOOde7r0nTBjVxvO+nZXVtfWNzcJWcXtnd2/fPThsaZkqTJpYMqk6IdKEUUGahhpGOokiiIeMtMPR1dRvPxClqRT3JktIwNFA0JhiZKzUd0s9HceIU5bBW8mpQAxeZ8IKWF/23bJX8WaAy8TPSRnkaPTdr14kccqJMJghrbu+l5hgjJShmJFJsZdqkiA8QgPStdSuIToYz56YwBOrRDCWypYwcKb+nhgjrnXGQ9vJkRnqRW8q/ud1UxNfBGMqktQQgeeL4pRBI+E0ERhRRbCxAUQUYUXtrRAPkULY2NyKNgR/8eVl0qpW/LNK7a5artfyOAqgBI7BKfDBOaiDG9AATYDBI3gGr+DNeXJenHfnY9664uQzR+APnM8fbKmX6Q==</latexit> Nominal Dynamics: <latexit sha1_base64="5zWECVxxAgRQ094V3Zvo64guhxM=">AAACM3icbZDLSgMxFIYz9VbrbdSlm2ARKoUyI0XdCFU3opsK9gJtKZn0tA3NXEgyhTrMO7nxRVwI4kIRt76D6bSIth4I/Hz/OTnJ7wScSWVZL0ZqYXFpeSW9mllb39jcMrd3qtIPBYUK9bkv6g6RwJkHFcUUh3oggLgOh5ozuBz7tSEIyXzvTo0CaLmk57Euo0Rp1Davo2ZySeTwEOL7djTI23GMz3DuPI8vbg7xnK9d7fxwAZ14mOC2mbUKVlJ4XthTkUXTKrfNp2bHp6ELnqKcSNmwrUC1IiIUoxziTDOUEBA6ID1oaOkRF2QrStbG+ECTDu76Qh9P4YT+noiIK+XIdXSnS1Rfznpj+J/XCFX3tBUxLwgVeHSyqBtyrHw8DhB3mACq+EgLQgXTb8W0TwShSsec0SHYs1+eF9Wjgn1cKN4Ws6XiNI402kP7KIdsdIJK6AqVUQVR9ICe0Rt6Nx6NV+PD+Jy0pozpzC76U8bXNxiBqzY=</latexit> zk+1 = (A + BK)zk + Bvk <latexit sha1_base64="vD1Khc06JGFABMJRR4soUu4lSZw=">AAACEnicbZDLSsNAFIZPvNZ6i7p0M1gE3ZREiroRCm5cVrAXaEOYTCft0MmFmYlYQ57Bja/ixoUibl25822cplnY1gMDP99/zpyZ34s5k8qyfoyl5ZXVtfXSRnlza3tn19zbb8koEYQ2ScQj0fGwpJyFtKmY4rQTC4oDj9O2N7qe+O17KiSLwjs1jqkT4EHIfEaw0sg1T9Nefknq8YRmj25qZRm6QjP0IaeuWbGqVl5oUdiFqEBRDdf87vUjkgQ0VIRjKbu2FSsnxUIxwmlW7iWSxpiM8IB2tQxxQKWT5nszdKxJH/mR0CdUKKd/J1IcSDkOPN0ZYDWU894E/ud1E+VfOikL40TRkEwX+QlHKkKTfFCfCUoUH2uBiWD6rYgMscBE6RTLOgR7/suLonVWtc+rtdtapV4r4ijBIRzBCdhwAXW4gQY0gcATvMAbvBvPxqvxYXxOW5eMYuYAZsr4+gX5ip7j</latexit> z0 = x0 <latexit sha1_base64="lropPlX/q8CN26bSOco3bh+u19c=">AAACAnicbVBNS8NAEJ3Ur1q/op7Ey2IRPJWkFBVPBRU8VrCt0Iay2WzapZtN2N0IoRQv/hUvHhTx6q/w5r9x2+agrXN6894MM+/5CWdKO863VVhaXlldK66XNja3tnfs3b2WilNJaJPEPJb3PlaUM0GbmmlO7xNJceRz2vaHlxO9/UClYrG401lCvQj3BQsZwdpQPfugq8IQR4xn6FrKWKKrTJiWqIueXXYqzrTQInBzUIa8Gj37qxvEJI2o0IRjpTquk2hvhKVmhNNxqZsqmmAyxH3aMdCcocobTS2M0bFhAhSaB8JYaDRlf2+McKRUFvlmMsJ6oOa1Cfmf1kl1eO6NmEhSTQWZHQpTjnSMJnmggElKtLEfMEwkM78iMsASE21SK5kQ3HnLi6BVrbinldpttVyv5XEU4RCO4ARcOIM63EADmkDgEZ7hFd6sJ+vFerc+ZqMFK9/Zhz9lff4A5U+XEQ==</latexit> Error Dynamics: <latexit sha1_base64="yXAyZJr/YBS8V9KSzlbXrHvNRj8=">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</latexit> ek+1 = Dk 2 6 6 4 w0 . . . wk 1 wk 3 7 7 5 <latexit sha1_base64="tlcCl1vLx+vNDhOa2EO0oL3Mfyo=">AAACAHicbVDLSsNAFJ34rPUVdeHCzWARXJVEiroRCm5cVrAPaEOYTG/boZNJmJkIIWTjr7hxoYhbP8Odf+M0zUJbD1w4nHPv3LkniDlT2nG+rZXVtfWNzcpWdXtnd2/fPjjsqCiRFNo04pHsBUQBZwLammkOvVgCCQMO3WB6O/O7jyAVi8SDTmPwQjIWbMQo0Uby7eNsUDyS0ZSIHPzMyXN8gx3frjl1pwBeJm5JaqhEy7e/BsOIJiEITTlRqu86sfYyIjWjHPLqIFEQEzolY+gbKkgIysuK3Tk+M8oQjyJpSmhcqL8nMhIqlYaB6QyJnqhFbyb+5/UTPbr2MibiRIOg80WjhGMd4VkaeMgkUM1TQwiVzPwV0wmRhGqTWdWE4C6evEw6F3X3st64b9SajTKOCjpBp+gcuegKNdEdaqE2oihHz+gVvVlP1ov1bn3MW1escuYI/YH1+QNo2pY7</latexit> e0 = 0 18
  16. Evolution of Decomposed of Dynamics: Nominal + Error <latexit sha1_base64="xbV+9pO3eQmtPm6HhKr83s7h9Mg=">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</latexit>

    Bc D† k " 1 n n X i=1 (A+BK)k+1x0+Ckv[0:k] +Dk b w(i) [0:k] ! <latexit sha1_base64="kBz8mEPNMEaIua+bkT/5zFdj6zw=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBZBEEoiRUUQCm5cVrQPaEOYTCft0MmDmYlYQn7FjQtF3Poj7vwbJ20W2npg4HDOvdwzx4s5k8qyvo3Syura+kZ5s7K1vbO7Z+5XOzJKBKFtEvFI9DwsKWchbSumOO3FguLA47TrTW5yv/tIhWRR+KCmMXUCPAqZzwhWWnLN6iDAaux56N5NJ6d2hq6uXbNm1a0Z0DKxC1KDAi3X/BoMI5IENFSEYyn7thUrJ8VCMcJpVhkkksaYTPCI9jUNcUClk86yZ+hYK0PkR0K/UKGZ+nsjxYGU08DTk3lSuejl4n9eP1H+pZOyME4UDcn8kJ9wpCKUF4GGTFCi+FQTTATTWREZY4GJ0nVVdAn24peXSeesbp/XG3eNWrNR1FGGQziCE7DhAppwCy1oA4EneIZXeDMy48V4Nz7moyWj2DmAPzA+fwBWapNM</latexit> S k+1 := <latexit sha1_base64="NTs81QWMntz8PzuMxJ8eqrkLvpw=">AAACBXicbVDLSsNAFJ34rPUVdamLwSIIQkmkqMuCG5cV7APaECaTm3bo5MHMRKghGzf+ihsXirj1H9z5N07TLLT1wIXDOffOnXu8hDOpLOvbWFpeWV1br2xUN7e2d3bNvf2OjFNBoU1jHoueRyRwFkFbMcWhlwggoceh642vp373HoRkcXSnJgk4IRlGLGCUKC255tHAB66Imw2KtzKPp5A/uNn4zM5z16xZdasAXiR2SWqoRMs1vwZ+TNMQIkU5kbJvW4lyMiIUoxzy6iCVkBA6JkPoaxqREKSTFZtzfKIVHwex0BUpXKi/JzISSjkJPd0ZEjWS895U/M/rpyq4cjIWJamCiM4WBSnHKsbTSLDPBFDFJ5oQKpj+K6YjIghVOriqDsGeP3mRdM7r9kW9cduoNRtlHBV0iI7RKbLRJWqiG9RCbUTRI3pGr+jNeDJejHfjY9a6ZJQzB+gPjM8fgVqZLg==</latexit> zk+1 <latexit sha1_base64="LTO2KvHs6v7QW747eENU4GWbPCQ=">AAACA3icbVDLSsNAFJ3UV62vqDvdBIvgqiRS1GXBjcsK9gFNCJPJbTt0MgkzE6GGgBt/xY0LRdz6E+78G6dpFtp64MLhnHvnzj1BwqhUtv1tVFZW19Y3qpu1re2d3T1z/6Ar41QQ6JCYxaIfYAmMcugoqhj0EwE4Chj0gsn1zO/dg5A05ndqmoAX4RGnQ0qw0pJvHrkhMIX9zC3eygKWQv7gZ5M898263bALWMvEKUkdlWj75pcbxiSNgCvCsJQDx06Ul2GhKGGQ19xUQoLJBI9goCnHEUgvK/bm1qlWQmsYC11cWYX6eyLDkZTTKNCdEVZjuejNxP+8QaqGV15GeZIq4GS+aJgyS8XWLBArpAKIYlNNMBFU/9UiYywwUTq2mg7BWTx5mXTPG85Fo3nbrLeaZRxVdIxO0Bly0CVqoRvURh1E0CN6Rq/ozXgyXox342PeWjHKmUP0B8bnD5ZYmL4=</latexit> zk <latexit sha1_base64="XVziT2hJ1zoQnfQSS8m7R6agyZc=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4sSRS1GXBjcsK9gFtCJPJTTt08mBmItSQjRt/xY0LRdz6D+78G6dpFtp64MLhnHvnzj1ewplUlvVtLC2vrK6tVzaqm1vbO7vm3n5Hxqmg0KYxj0XPIxI4i6CtmOLQSwSQ0OPQ9cbXU797D0KyOLpTkwSckAwjFjBKlJZc82jgA1fEzQbFW5nHU8gf3Gx8Zue5a9asulUALxK7JDVUouWaXwM/pmkIkaKcSNm3rUQ5GRGKUQ55dZBKSAgdkyH0NY1ICNLJis05PtGKj4NY6IoULtTfExkJpZyEnu4MiRrJeW8q/uf1UxVcORmLklRBRGeLgpRjFeNpJNhnAqjiE00IFUz/FdMREYQqHVxVh2DPn7xIOud1+6LeuG3Umo0yjgo6RMfoFNnoEjXRDWqhNqLoET2jV/RmPBkvxrvxMWtdMsqZA/QHxucPhGiZMA==</latexit> zk 1 <latexit sha1_base64="UkycVXA30mQJf6VQNB3eE1NKQDU=">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</latexit> Bc D† k 2 " 1 n n X i=1 b e(i) k 1 ! <latexit sha1_base64="q/IhJOpIWAi3U6KAgLuJT2XnhXM=">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</latexit> Bc D† k 1 " 1 n n X i=1 b e(i) k ! <latexit sha1_base64="xLlJDwZXaBurwMn4iNP+I3bVzMs=">AAACeXicbVFNbxMxEPUuXyV8hXKEgyEqSmkVZWkEXJAq4MCxSKStFCerWe/sxorXu7K9RZHl/8Bv48Yf4cIFb5oDtIxk6+nNm/H4TdZIYex4/DOKb9y8dfvOzt3evfsPHj7qP949NXWrOU55LWt9noFBKRROrbASzxuNUGUSz7LVxy5/doHaiFp9tesG5xWUShSCgw1U2v/OKrDLLKMfUscuQGNjhKyVXzhOGRc63J2Ag6SfUrfyC5ZDWaL2lEks7JAVGrhLvFOBMW2VOvE+8QtF2SHLUVoIbQ/ZZk7H16A8+yZyXIJ16EO/g6B1Q7HvqWdalEu7n/YH49F4E/Q6SLZgQLZxkvZ/sLzmbYXKcgnGzJJxY+cOtBVcou+x1mADfAUlzgJUUKGZu81Enu4FJqdFrcNRlm7YvyscVMasqywoOxvM1VxH/i83a23xbu6EalqLil8+VLSS2pp2a6C50MitXAcAXIswK+VLCF7asKxeMCG5+uXr4PT1KHkzmnyZDI4nWzt2yFPyggxJQt6SY/KZnJAp4eRX9Czai15Gv+Pn8TB+dSmNo23NE/JPxEd/ADCCwiE=</latexit> Bc D† k " 1 n n X i=1 b e(i) k+1 ! <latexit sha1_base64="zfS8EbbvEUlUfyZcdCrSiGPalok=">AAACIXicbVDLSsNAFJ34rPVVdelmsAqCUJJStMuCG3cqtCo0pUymN3Xo5MHMjVhCfsWNv+LGhSLuxJ9xmmahrQcGDufew7lzvFgKjbb9ZS0sLi2vrJbWyusbm1vblZ3dGx0likOHRzJSdx7TIEUIHRQo4S5WwAJPwq03Op/Mbx9AaRGFbRzH0AvYMBS+4AyN1K80Xe37LBByTC/b1PjEMBE4phqQRj49TN08I/VkAtljPx2dOFl2SIXuV6p2zc5B54lTkCopcNWvfLqDiCcBhMgl07rr2DH2UqZQcAlZ2U00xIyP2BC6hoYsAN1L8/SMHhllQP1ImRcizdXfjpQFWo8Dz2wGDO/17Gwi/jfrJug3e6kI4wQh5NMgP5EUIzqpiw6EAo6mnYFgXAlzK+X3TDGOptSyKcGZ/fI8uanXnNNa47pebTWKOkpknxyQY+KQM9IiF+SKdAgnT+SFvJF369l6tT6sz+nqglV49sgfWN8/dBOjpw==</latexit> OT ambiguity set of xk+1 is <latexit sha1_base64="xq0ID4IB6qYVdextWQznJzIrNVo=">AAACE3icbVBNSwMxEM36bf2qevQSrIIolN1S1KPgxWMFq0Jbymw6W0OT7JJkhWXpf/DiX/HiQRGvXrz5b0zbPWj1wcDjvZlM5oWJ4Mb6/pc3Mzs3v7C4tFxaWV1b3yhvbl2bONUMmywWsb4NwaDgCpuWW4G3iUaQocCbcHA+8m/uURseqyubJdiR0Fc84gysk7rlw7aJIpBcZNSATAQaGkd0rz1+OWcZqCF288FRMNzrlit+1R+D/iVBQSqkQKNb/mz3YpZKVJYJMKYV+Int5KAtZwKHpXZqMAE2gD62HFUg0XTy8eYh3XdKj0axdqUsHas/J3KQxmQydJ0S7J2Z9kbif14rtdFpJ+cqSS0qNlkUpYLamI4Coj2ukVmXR48D09z9lbI70MCsi7HkQgimT/5LrmvV4Lhav6xVzupFHEtkh+ySAxKQE3JGLkiDNAkjD+SJvJBX79F79t6890nrjFfMbJNf8D6+AdIFnhI=</latexit> samples of ek+1 <latexit sha1_base64="XBJeVXh74MNo2oQZrAlb8mpsDDU=">AAACA3icbVDLSgMxFM34rPU16k43wSK4KjOlqMuCG1dSwT6gHUomk2lD8xiSjDAMBTf+ihsXirj1J9z5N6btLLT1QOBwzr3cnBMmjGrjed/Oyura+sZmaau8vbO7t+8eHLa1TBUmLSyZVN0QacKoIC1DDSPdRBHEQ0Y64fh66nceiNJUinuTJSTgaChoTDEyVhq4x30dx4hTlsFbyalADEaZsALWA7fiVb0Z4DLxC1IBBZoD96sfSZxyIgxmSOue7yUmyJEyFDMyKfdTTRKEx2hIepbaK0QH+SzDBJ5ZJYKxVPYJA2fq740cca0zHtpJjsxIL3pT8T+vl5r4KsipSFJDBJ4filMGjYTTQmBEFcHG5o8oworav0I8QgphY2sr2xL8xcjLpF2r+hfV+l2t0qgXdZTACTgF58AHl6ABbkATtAAGj+AZvII358l5cd6dj/noilPsHIE/cD5/ABbvl8U=</latexit> Nominal dynamics <latexit sha1_base64="hag3XYe/uNNnX6Q+hKf9w1yAGMQ=">AAAB8nicbVDLSgMxFM3UV62vqks3wSIIQpkpRV0W3LisYB8wHUomk2lDk8yQ3BHK0M9w40IRt36NO//GtJ2Fth4IHM65l9xzwlRwA6777ZQ2Nre2d8q7lb39g8Oj6vFJ1ySZpqxDE5HofkgME1yxDnAQrJ9qRmQoWC+c3M393hPThifqEaYpCyQZKR5zSsBK/sDEMZFcTPHVsFpz6+4CeJ14BamhAu1h9WsQJTSTTAEVxBjfc1MIcqKBU8FmlUFmWErohIyYb6kikpkgX5w8wxdWiXCcaPsU4IX6eyMn0pipDO2kJDA2q95c/M/zM4hvg5yrNAOm6PKjOBMYEjzPjyOuGQUbOOKEam5vxXRMNKFgW6rYErzVyOuk26h71/XmQ6PWahZ1lNEZOkeXyEM3qIXuURt1EEUJekav6M0B58V5dz6WoyWn2DlFf+B8/gDJIpDj</latexit> + <latexit sha1_base64="mB6exER8nKgUgTRKK+dNqM98Jj4=">AAACCXicbVDLSsNAFJ3UV62vqEs3g0VwVZJS1GVBBHdW6AvaUCaTSTt0JgkzN0Io3brxV9y4UMStf+DOv3HaZqGtZ3U45x7uvcdPBNfgON9WYW19Y3OruF3a2d3bP7APj9o6ThVlLRqLWHV9opngEWsBB8G6iWJE+oJ1/PH1zO88MKV5HDUhS5gnyTDiIacEjDSwcV+HIZFcZPhGqVjhuyY2aT5MOWRYMxjYZafizIFXiZuTMsrRGNhf/SCmqWQRUEG07rlOAt6EKOBUsGmpn2qWEDomQ9YzNCKSaW8y/2SKz4wS4NDcEcYR4Ln6OzEhUutM+mZSEhjpZW8m/uf1UgivvAmPkhRYRBeLwlRgiPGsFhxwxSiYFgJOqOLmVkxHRBEKprySKcFdfnmVtKsV96JSu6+W67W8jiI6QafoHLnoEtXRLWqgFqLoET2jV/RmPVkv1rv1sRgtWHnmGP2B9fkDBpyZ3w==</latexit> Error OT ambiguity set 19 <latexit sha1_base64="uK5fLAx/2d86jkqVBXUrLPTy2TY=">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</latexit> zk+1 <latexit sha1_base64="0g7ljIyA8P3SyJRFybDY9amtWGw=">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</latexit> ek+1 <latexit sha1_base64="2gNVHEsPbLQhErTs9RmehnfCTys=">AAAB/nicbVDLSsNAFJ3UV62vqLhyEyyCIJREirosuHFZwT6gDWEyvW2HTiZhZiLUIeCvuHGhiFu/w51/4zTNQlsPXDicc+/cuSdMGJXKdb+t0srq2vpGebOytb2zu2fvH7RlnAoCLRKzWHRDLIFRDi1FFYNuIgBHIYNOOLmZ+Z0HEJLG/F5NE/AjPOJ0SAlWRgrsI93PH9EhSyF7DPTk3MuywK66NTeHs0y8glRRgWZgf/UHMUkj4IowLGXPcxPlaywUJQyySj+VkGAywSPoGcpxBNLX+ebMOTXKwBnGwhRXTq7+ntA4knIahaYzwmosF72Z+J/XS9Xw2teUJ6kCTuaLhilzVOzMsnAGVABRbGoIJoKavzpkjAUmyiRWMSF4iycvk/ZFzbus1e/q1Ua9iKOMjtEJOkMeukINdIuaqIUI0ugZvaI368l6sd6tj3lrySpmDtEfWJ8/B76WIw==</latexit> zk+1 <latexit sha1_base64="SKxR0s2QFbc06BjBN2/w5g64Wy8=">AAACG3icbVBNSwMxEM36bf2qevQSrIIilN0i6lHw4lHBqtCtZTY7a0Oz2SXJKiXs//DiX/HiQRFPggf/jWntwa8HA4/3ZpKZF+WCa+P7H97Y+MTk1PTMbGVufmFxqbq8cq6zQjFsskxk6jICjYJLbBpuBF7mCiGNBF5EvaOBf3GDSvNMnpl+ju0UriVPOAPjpE61EeokgZSLPt2w4fA9y/ogy/CWx9gFY7Hs2N5OUF7ZLb5dlhu0U635dX8I+pcEI1IjI5x0qm9hnLEiRWmYAK1bgZ+btgVlOBNYVsJCYw6sB9fYclRCirpth7uUdNMpMU0y5UoaOlS/T1hIte6nketMwXT1b28g/ue1CpMctC2XeWFQsq+PkkJQk9FBUDTmCplxucQcmOJuV8q6oIAZF2fFhRD8PvkvOW/Ug7367mmjdrg7imOGrJF1skUCsk8OyTE5IU3CyB15IE/k2bv3Hr0X7/WrdcwbzaySH/DePwGpu6G6</latexit> b e(i) k+1 <latexit sha1_base64="3RoHHu675qYqLgNkuwEbW/QbGPA=">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</latexit> ) <latexit 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  17. Wasserstein Tube MPC <latexit sha1_base64="Vij8UI0HFzmD+1C8H7FFhW2fdWI=">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</latexit> min PN 1 t=0 kzk|t

    k2 Q + kvk|t k2 R s.t. vk|t , zk|t zk+1|t = (A + BK)zk|t + Bvk|t Kzk|t + vk|t 2 U KEk supQ2S k CVaRQ maxj a> j xk|t + bj  0 zN|t 2 Zf z0|t = xt <latexit sha1_base64="JXpVC6/bjxnLl8ip04V0VSyDqDw=">AAACA3icbVC7SgNBFJ2Nrxhfq3baDAbBxrAbgopVII2NECEvyC5hdjKbDJl9MHNXDEvAxl+xsVDE1p+w82+cJFto4oGBwzn3cuccLxZcgWV9G7mV1bX1jfxmYWt7Z3fP3D9oqSiRlDVpJCLZ8YhigoesCRwE68SSkcATrO2NalO/fc+k4lHYgHHM3IAMQu5zSkBLPfPIUb5PAi7G2AH2AJ6fthvnt/Xa9aRnFq2SNQNeJnZGiihDvWd+Of2IJgELgQqiVNe2YnBTIoFTwSYFJ1EsJnREBqyraUgCptx0lmGCT7XSx34k9QsBz9TfGykJlBoHnp4MCAzVojcV//O6CfhXbsrDOAEW0vkhPxEYIjwtBPe5ZBR0/j4nVHL9V0yHRBIKuraCLsFejLxMWuWSfVGq3JWL1UpWRx4doxN0hmx0iaroBtVRE1H0iJ7RK3oznowX4934mI/mjGznEP2B8fkDaUyXUg==</latexit> WT-MPC: <latexit sha1_base64="iwxdigCnHss2VSRVSrZFmScqqac=">AAACD3icbVDLSgMxFM34rPVVdekmWBRXZaYUdVlw47KifUBbSiZzpw3NY0gyxVL6B278FTcuFHHr1p1/Y/pYaOuBwOGce5OcEyacGev7397K6tr6xmZmK7u9s7u3nzs4rBmVagpVqrjSjZAY4ExC1TLLoZFoICLkUA/71xO/PgBtmJL3dphAW5CuZDGjxDqpkztrmTgmgvEhvgOLVYypkgN4wBFY0IJJ9wVGcSeX9wv+FHiZBHOSR3NUOrmvVqRoKkBayokxzcBPbHtEtLuNwzjbSg0khPZJF5qOSiLAtEfTPGN86pQIx0q7Iy2eqr83RkQYMxShmxTE9syiNxH/85qpja/aIyaT1IKks4filGOr8KQcHDEN1LouIkaoZpPktEc0oa4Lk3UlBIuRl0mtWAguCqXbYr5cmteRQcfoBJ2jAF2iMrpBFVRFFD2iZ/SK3rwn78V79z5moyvefOcI/YH3+QPVvpx/</latexit> Set of convex deterministic <latexit sha1_base64="4VIZ/Xc7VtAyi6GVzxa07wmx5IE=">AAACGXicbVDLTgIxFO34RHyhLt00gokrMkOIuiRx4xITeSRASKd0oKHTTto7JjjOb7jxV9y40BiXuvJvLAMLBU9yk5Nz7u3tPX4kuAHX/XZWVtfWNzZzW/ntnd29/cLBYdOoWFPWoEoo3faJYYJL1gAOgrUjzUjoC9byx1dTv3XHtOFK3sIkYr2QDCUPOCVgpX7B7ZogICEXE0yVNKAJl2CwkriUdLPnE1/ELL3vJ+MHSNNSv1B0y24GvEy8OSmiOer9wmd3oGgcMglUEGM6nhtBLyEaOBUszXdjwyJCx2TIOpZKEjLTS7LVKT61ygAHStuSgDP190RCQmMmoW87QwIjs+hNxf+8TgzBZS/hMoqBSTpbFMQCg8LTmPCAa0bBpjLghGpu/4rpiGhCwYaZtyF4iycvk2al7J2XqzeVYq06jyOHjtEJOkMeukA1dI3qqIEoekTP6BW9OU/Oi/PufMxaV5z5zBH6A+frB1Crobc=</latexit> constraints on zk|t th Theorem (informal). • the optimal control problem is efficiently solvable, • the receding-horizon implementation is recursively feasible 
 under appropriate assumptions on & constraint tightening, <latexit sha1_base64="BE+kcQivDBeDMiB8bUHPOTssb6U=">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</latexit> Zf <latexit sha1_base64="4H3bNdlB/cm8GVulqUMb7U42Eo0=">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</latexit> 20 • … and closed-loop stability is still open ☺︎
  18. Summary <latexit sha1_base64="qpI8/ISO+KqNL6uAXeH4IIoOvII=">AAACRXicbZDNa1NBFMXn1a8aPxp16WZoELoI4b0i1mVLXbgQjGDaQl4I903uSy6dj8fMfaXxkX/OjXt3/gduXFiKW52kWWjrgYHDOffOML+i0hQ4Tb8lG7du37l7b/N+68HDR4+32k+eHgVXe4UD5bTzJwUE1GRxwMQaTyqPYAqNx8Xp4bI/PkMfyNmPPK9wZGBqqSQFHKNxO89DWYIhPZc54zkXZXPgNHhipK58B/YTMkd3OEPblW/ybedLjX7RlQOr0DOQ5bnse1fBdHWlPCOQ7ysmA3rc7qS9dCV502Rr0xFr9cftr/nEqdqgZaUhhGGWVjxqwDMpjYtWXgesQJ3CFIfRWjAYRs2KwkK+iMlEls7HY1mu0r83GjAhzE0RJw3wLFzvluH/umHN5etRQ7aqGa26eqistWQnl0jlhDwqjgQnBCqSIyXVDDwojuBbEUJ2/cs3zdFuL3vVe/lht7O/s8axKZ6LbbEjMrEn9sVb0RcDocRn8V38FBfJl+RHcpn8uhrdSNY7z8Q/Sn7/AcaOslc=</latexit> Aolaritei, Lanzetti, Chen, D¨ orfler, Uncertainty Propagation

    via Optimal <latexit sha1_base64="R2BRef5byOTkmqPY0av6fxDx/ZA=">AAACEXicbVA9SwNBEN3zM8avqKXNYhBSSLg7RC0jNpYR8wXJEfY2e8mS3b1jd04IIX/Bxr9iY6GIrZ2d/8ZNcoUmPhh4vDfDzLwwEdyA6347K6tr6xubua389s7u3n7h4LBh4lRTVqexiHUrJIYJrlgdOAjWSjQjMhSsGQ5vpn7zgWnDY1WDUcICSfqKR5wSsFK3UOqYKCKSixGuaaJMEmvA1zLk/ZTDCN8zMGfYd32/Wyi6ZXcGvEy8jBRRhmq38NXpxTSVTAEVxJi25yYQjIkGTgWb5DupYQmhQ9JnbUsVkcwE49lHE3xqlR6OYm1LAZ6pvyfGRBozkqHtlAQGZtGbiv957RSiq2DMVZICU3S+KEoFhhhP48E9rhkFm0aPE6q5vRXTAdGEgg0xb0PwFl9eJg2/7F2Uz+/8YqWUxZFDx+gElZCHLlEF3aIqqiOKHtEzekVvzpPz4rw7H/PWFSebOUJ/4Hz+ABC1nGg=</latexit> Transport Ambiguity Sets, 2022 <latexit sha1_base64="tJOx0YRHb559M9RPYORPERkgICk=">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</latexit> Aolaritei, Lanzetti, D¨ orfler, Capture, Propagate, and Control Distributional <latexit sha1_base64="gu+WMtbEXi4BnExUGl8SR/6GYX8=">AAACBHicbVA9SwNBEN2LXzF+RS3TLAYhhYS7KGoZsLGM4CWBJIS9zV6yZHfv2J0TjiOFjX/FxkIRW3+Enf/GzUehiQ8GHu/NMDMviAU34LrfTm5tfWNzK79d2Nnd2z8oHh41TZRoynwaiUi3A2KY4Ir5wEGwdqwZkYFgrWB8M/VbD0wbHql7SGPWk2SoeMgpASv1i6WuCUMiuUixryjTQLiC9AzX3Np5v1h2q+4MeJV4C1JGCzT6xa/uIKKJZAqoIMZ0PDeGXkY0cCrYpNBNDIsJHZMh61iqiGSml82emOBTqwxwGGlbCvBM/T2REWlMKgPbKQmMzLI3Ff/zOgmE172MqzgBpuh8UZgIDBGeJoIHXDMKNoABJ1RzeyumI6IJBZtbwYbgLb+8Spq1qndZvbirleuVRRx5VEInqII8dIXq6BY1kI8oekTP6BW9OU/Oi/PufMxbc85i5hj9gfP5A1RNlyY=</latexit> Uncertainty, 2023 <latexit sha1_base64="5veQZyOQaGmdOV5P86G3pvYfop0=">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</latexit> Shafieezadeh-Abadeh, Aolaritei, D¨ orfler, Kuhn, New Perspectives on <latexit sha1_base64="HJAmmZ/jTeQVfGtE77PbAv01blc=">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</latexit> Regularization and Computation in Optimal Transport-Based Distributionally <latexit sha1_base64="K5XHzZVqI9x6odPxugNxoiUbeOs=">AAACBnicbVA9SwNBEN3zM8avqKUIi0FIFe6CqGXAxs4o5gOSI+zt7SVLdveO3TkhhlQ2/hUbC0Vs/Q12/hs3yRWa+GDg8d4MM/OCRHADrvvtLC2vrK6t5zbym1vbO7uFvf2GiVNNWZ3GItatgBgmuGJ14CBYK9GMyECwZjC4nPjNe6YNj9UdDBPmS9JTPOKUgJW6haOOiSIiuRji2zhIDeDrBLjkD5lfdMvuFHiReBkpogy1buGrE8Y0lUwBFcSYtucm4I+IBk4FG+c7qWEJoQPSY21LFZHM+KPpG2N8YpUQR7G2pQBP1d8TIyKNGcrAdkoCfTPvTcT/vHYK0YU/4ipJgSk6WxSlAkOMJ5ngkGtGwUYQckI1t7di2ieaULDJ5W0I3vzLi6RRKXtn5dObSrFayuLIoUN0jErIQ+eoiq5QDdURRY/oGb2iN+fJeXHenY9Z65KTzRygP3A+fwC425k3</latexit> Robust Optimization<latexit sha1_base64="1m2gDdCaecg1iNZX+BVQl3y5lU4=">AAAB+XicbVBNS8NAEJ3Ur1q/oh69LBahBylJFPVY8OKxgv2ANpTNZtMu3U3C7qZQQv+JFw+KePWfePPfuG1z0NYHA4/3ZpiZF6ScKe0431ZpY3Nre6e8W9nbPzg8so9P2irJJKEtkvBEdgOsKGcxbWmmOe2mkmIRcNoJxvdzvzOhUrEkftLTlPoCD2MWMYK1kQa23VdRhAXjU3SJPMe7GthVp+4sgNaJW5AqFGgO7K9+mJBM0FgTjpXquU6q/RxLzQins0o/UzTFZIyHtGdojAVVfr64fIYujBKiKJGmYo0W6u+JHAulpiIwnQLrkVr15uJ/Xi/T0Z2fszjNNI3JclGUcaQTNI8BhUxSos3XIcNEMnMrIiMsMdEmrIoJwV19eZ20vbp7U79+9KqNWhFHGc7gHGrgwi004AGa0AICE3iGV3izcuvFerc+lq0lq5g5hT+wPn8Aa8qSIg==</latexit> , 2023 <latexit sha1_base64="mAOJgSYhbAT5F3jmzMI4zPFeg8U=">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</latexit> Aolaritei, Fochesato, Lygeros, D¨ orfler, Wasserstein Tube MPC with Exact <latexit sha1_base64="IGA1MKtqzsT31CNsapO/k4vc+0g=">AAACEHicbVC7SgNBFJ2NrxhfUUubwSCmkLC7iloGbCwjmAckIdydzCZDZmeXmbtCCPkEG3/FxkIRW0s7/8bJo9DEAwOHc87lzj1BIoVB1/12Miura+sb2c3c1vbO7l5+/6Bm4lQzXmWxjHUjAMOlULyKAiVvJJpDFEheDwY3E7/+wLURsbrHYcLbEfSUCAUDtFInf9oyYQiRkENaVYxrBKFwSCs6TqA3zZxR3/XPO/mCW3KnoMvEm5MCmaPSyX+1ujFLI66QSTCm6bkJtkegUTDJx7lWangCbAA93rRUQcRNezQ9aExPrNKlYaztU0in6u+JEUTGDKPAJiPAvln0JuJ/XjPF8Lo9EipJkSs2WxSmkmJMJ+3QrtCcoS2jK4BpYf9KWR80MLQd5mwJ3uLJy6Tml7zL0sWdXygX53VkyRE5JkXikStSJrekQqqEkUfyTF7Jm/PkvDjvzscsmnHmM4fkD5zPH7TknEI=</latexit> Uncertainty Propagation, 2023 • pitfalls of robust & stochastic uncertainty descriptions • OT ambiguity capturing of distributional uncertainty • propagation results in yet another OT ambiguity set • today’s application to control: Wasserstein tube MPC 22 <latexit sha1_base64="/+FRpYVTL/Q5DmLt2ZGqfvDazEI=">AAACAnicbVDLSsNAFL3xWesr6krcDBbBVUmkqMuCG5cV7AOaUCbTaTt0MgkzEyGE4sZfceNCEbd+hTv/xkmahbYeGDicc19zgpgzpR3n21pZXVvf2KxsVbd3dvf27YPDjooSSWibRDySvQArypmgbc00p71YUhwGnHaD6U3udx+oVCwS9zqNqR/isWAjRrA20sA+zrxiSEZSLGZeiPUkCFBrIGYDu+bUnQJombglqUGJ1sD+8oYRSUIqNOFYqb7rxNrPsNSMcDqreomiMSZTPKZ9QwUOqfKzYvsMnRlliEaRNE9oVKi/OzIcKpWGganMb1SLXi7+5/UTPbr2MybiRFNB5otGCUc6QnkeaMgkJZqnhmAimbkVkQmWmGiTWtWE4C5+eZl0LuruZb1x16g1G2UcFTiBUzgHF66gCbfQgjYQeIRneIU368l6sd6tj3npilX2HMEfWJ8/8cqXvQ==</latexit> P n <latexit sha1_base64="k1AtdIimUWBqygfvFGKKOZEHPFk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE8lUSKeix48VjBfkAaymY7aZdudsPupFBCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXpYIb9LxvZ2Nza3tnt7RX3j84PDqunJy2jco0gxZTQuluRA0ILqGFHAV0Uw00iQR0ovH93O9MQBuu5BNOUwgTOpQ85oyilYLehGpIDRdK9itVr+Yt4K4TvyBVUqDZr3z1BoplCUhkghoT+F6KYU41ciZgVu5lBlLKxnQIgaWSJmDCfHHyzL20ysCNlbYl0V2ovydymhgzTSLbmVAcmVVvLv7nBRnGd2HOZZohSLZcFGfCReXO/3cHXANDMbWEMs3trS4bUU0Z2pTKNgR/9eV10r6u+Te1+mO92qgXcZTIObkgV8Qnt6RBHkiTtAgjijyTV/LmoPPivDsfy9YNp5g5I3/gfP4AuIeRgg==</latexit> " <latexit sha1_base64="GUD9jfdS/NnObjoPykcSdja/fbk=">AAACIHicbVDNS8MwHE39nPNr6tFLcAjzMloZTgRh6MXjBPcBay1plm1haVqSdFBK/xQv/itePCiiN/1rTLcKuvkg8Hjv/ZJfnhcyKpVpfhpLyyura+uFjeLm1vbObmlvvy2DSGDSwgELRNdDkjDKSUtRxUg3FAT5HiMdb3yd+Z0JEZIG/E7FIXF8NOR0QDFSWnJLddtHauR58MpN7AkSJJSUBTy9x5XEnl6f4Bjx9CfWdHl6Ai8u3VLZrJpTwEVi5aQMcjTd0ofdD3DkE64wQ1L2LDNUToKEopiRtGhHkoQIj9GQ9DTlyCfSSaYbpPBYK304CIQ+XMGp+nsiQb6Use/pZLannPcy8T+vF6nBuZNQHkaKcDx7aBAxqAKYtQX7VBCsWKwJwoLqXSEeIYGw0p0WdQnW/JcXSfu0ap1Va7e1cqOW11EAh+AIVIAF6qABbkATtAAGD+AJvIBX49F4Nt6M91l0ychnDsAfGF/fd8mjtA==</latexit> Bc " (P n) := <latexit sha1_base64="PCs2Ar7mF03MsgdPaaZz5xYetx4=">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</latexit> f# Bc " (P n) = Bc f 1 " (f# P n) <latexit sha1_base64="k1AtdIimUWBqygfvFGKKOZEHPFk=">AAAB8nicbVBNS8NAEN34WetX1aOXYBE8lUSKeix48VjBfkAaymY7aZdudsPupFBCf4YXD4p49dd489+4bXPQ1gcDj/dmmJkXpYIb9LxvZ2Nza3tnt7RX3j84PDqunJy2jco0gxZTQuluRA0ILqGFHAV0Uw00iQR0ovH93O9MQBuu5BNOUwgTOpQ85oyilYLehGpIDRdK9itVr+Yt4K4TvyBVUqDZr3z1BoplCUhkghoT+F6KYU41ciZgVu5lBlLKxnQIgaWSJmDCfHHyzL20ysCNlbYl0V2ovydymhgzTSLbmVAcmVVvLv7nBRnGd2HOZZohSLZcFGfCReXO/3cHXANDMbWEMs3trS4bUU0Z2pTKNgR/9eV10r6u+Te1+mO92qgXcZTIObkgV8Qnt6RBHkiTtAgjijyTV/LmoPPivDsfy9YNp5g5I3/gfP4AuIeRgg==</latexit> " <latexit sha1_base64="sjlPuQ12qw6GUmCDlxYS0CfJcIU=">AAAB/HicbVDLSsNAFL3xWesr2qWbwSK4KokUdVlw47KCfUAbwmQ6bYdOJmFmIsQQf8WNC0Xc+iHu/BunaRbaeuDC4Zx75849QcyZ0o7zba2tb2xubVd2qrt7+weH9tFxV0WJJLRDIh7JfoAV5UzQjmaa034sKQ4DTnvB7Gbu9x6oVCwS9zqNqRfiiWBjRrA2km/XsmxYvJKRFIv80Xfz3LfrTsMpgFaJW5I6lGj79tdwFJEkpEITjpUauE6svQxLzQineXWYKBpjMsMTOjBU4JAqLyvW5ujMKCM0jqQpoVGh/p7IcKhUGgamM8R6qpa9ufifN0j0+NrLmIgTTQVZLBonHOkIzZNAIyYp0Tw1BBPJzF8RmWKJiTZ5VU0I7vLJq6R70XAvG827Zr3VLOOowAmcwjm4cAUtuIU2dIBACs/wCm/Wk/VivVsfi9Y1q5ypwR9Ynz/PZJV8</latexit> z1 <latexit sha1_base64="r1rPK4D6ooSH9Lkti/dEyALfk9o=">AAAB/HicbVDLSsNAFL2pr1pf0S7dBIvgqiSlqMuCG5cV7APaECbTSTt0MgkzEyGG+CtuXCji1g9x5984TbPQ1gMXDufcO3fu8WNGpbLtb6Oysbm1vVPdre3tHxwemccnfRklApMejlgkhj6ShFFOeooqRoaxICj0GRn485uFP3ggQtKI36s0Jm6IppwGFCOlJc+sZ9m4eCXDKeL5o9fKc89s2E27gLVOnJI0oETXM7/GkwgnIeEKMyTlyLFj5WZIKIoZyWvjRJIY4TmakpGmHIVEulmxNrfOtTKxgkjo4soq1N8TGQqlTENfd4ZIzeSqtxD/80aJCq7djPI4UYTj5aIgYZaKrEUS1oQKghVLNUFYUP1XC8+QQFjpvGo6BGf15HXSbzWdy2b7rt3otMs4qnAKZ3ABDlxBB26hCz3AkMIzvMKb8WS8GO/Gx7K1YpQzdfgD4/MH0OqVfQ==</latexit> z2 <latexit sha1_base64="Xpi8enVimMC1uI6NqNsyZodaZlQ=">AAAB/HicbVDLSsNAFL3xWesr2qWbwSK4KokUdVlw47KCfUAbwmQ6bYdOJmFmIsQQf8WNC0Xc+iHu/BunaRbaeuDC4Zx75849QcyZ0o7zba2tb2xubVd2qrt7+weH9tFxV0WJJLRDIh7JfoAV5UzQjmaa034sKQ4DTnvB7Gbu9x6oVCwS9zqNqRfiiWBjRrA2km/XsmxYvJKRFIv80Rd57tt1p+EUQKvELUkdSrR9+2s4ikgSUqEJx0oNXCfWXoalZoTTvDpMFI0xmeEJHRgqcEiVlxVrc3RmlBEaR9KU0KhQf09kOFQqDQPTGWI9VcveXPzPGyR6fO1lTMSJpoIsFo0TjnSE5kmgEZOUaJ4agolk5q+ITLHERJu8qiYEd/nkVdK9aLiXjeZds95qlnFU4ARO4RxcuIIW3EIbOkAghWd4hTfryXqx3q2PReuaVc7U4A+szx8sYZW5</latexit> zn <latexit sha1_base64="kPiweoemMIy1foxf2AT3pE7hZjM=">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</latexit> sample distribution P n <latexit sha1_base64="RI3SrSWrgNIM/oUPTZsCun1gEM0=">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</latexit> true distribution P ? <latexit sha1_base64="N/vn0U4LNSeUxJfvcvfYcLYsd7o=">AAAD73icdVLLbtNAFJ3GPEp4pbBBYjMiqsQCRXYlHsvyEGJZBGkrxVY0Hl8no87DnRmnNSPzG+wQCzYs4Bf4Dv6GcZIinKRXGvno3HOfvmnBmbFh+GerE1y5eu369o3uzVu379zt7dw7NKrUFIZUcaWPU2KAMwlDyyyH40IDESmHo/TkdeM/moE2TMmPtiogEWQiWc4osZ4a9x7EJs+JYLzC2EexSclshQ3Yca8fDsK54XUQLUEfLe1gvNP5HWeKlgKkpZwYM4rCwiaOaMsoh7oblwYKQk/IBEYeSiLAPMlmrDBzmLj5NDXe9c4M50r7Jy2es/8HOyKMqUTqlYLYqVn1NeQm36i0+YvEMVmUFiRdFMpLjq3CzWpwxjRQ6zeRMUI1821jOiWaUOsX2KpyRkzlW2jN5JqCVilu6s0dbZ6hTVNzWioL6ymaVZj1etrkG9hMrWqzeYI2d5770epudxfP/NiqGfEN+D+n4YPvTPG3PsKlHmS1G9YXSNRO1huUL3kxJSlYFzcdLMWLTzeWcEaVEERmLjZMFBzO61GUOJ+GWzJ2/aheUTUtLST/0l2iUlpJvzCvHSULxkX1ZSmV/gRatdXhhdqffLR64OvgcG8QPRs8fb/X33+1PP5t9BA9Qo9RhJ6jffQOHaAhougz+o5+ol/BafAl+Bp8W0g7W8uY+6hlwY+/T7FZNg==</latexit> ambiguity set <latexit sha1_base64="27Z3oOrFCnQdYK+XpniGpM8NSIw=">AAAD9HicdVLLjtMwFPU0PIby6sASIUVUI7FAVTLitRweQiyLoDMjNVHkODdTaxw72E5niuUVv8EOsWDDAj6A7+BvcJIOIm3HkpWjc4/vPffmpiWjSgfBn62ed+nylavb1/rXb9y8dXuwc+dAiUoSmBDBhDxKsQJGOUw01QyOSgm4SBkcpiev6vjhHKSign/QixLiAh9zmlOCtaOSwf08iYYmahIZssDcRgXWszT1xwm3yWAYjILm+OsgXIIhWp5xstP7HWWCVAVwTRhWahoGpY4NlpoSBrYfVQpKTE7wMUwd5LgA9Sib01I1MDaNE+vvumDm50K6y7XfsP8/NrhQalGkTln7VauxmtwUm1Y6fx4bystKAydtobxivhZ+PR8/oxKIZgsHMJHU2fbJDEtMtJtip8opVgtnodOTqQtqIZiymx1t7qFLE/WxEhrWU9SjUOv1pMo3sJlY1WZNgi53lrvWbL+/689d26Ju8TW4PyfhvXMm2Bv3wqQOZNZM7DkqrOF2g/IFK2c4BW2WS9SI208/4nBKRFFgnplI0aJkcGanYWxcGqZxYoahXVHVllrJv3QXqIQU3A3Maadxy5jQXpRSyE8gRVcdnKvdyoerC74ODvZG4dPRk3ePh/svl8u/je6hB+ghCtEztI/eojGaIII+o+/oJ/rlzb0v3lfvWyvtbS3f3EWd4/34C95WW50=</latexit> f# P n Conclusions: better uncertainty model • parametrizable from data • strong statistical properties • efficiently computable • expressive: space & likelihood • closure under propagation • robust to distribution shift