Clustering <latexit 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Generative modeling <latexit 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Dimensionality reduction <latexit 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Supervised learning: <latexit 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Regression <latexit 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? <latexit 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=) <latexit 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Classification <latexit 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? <latexit 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=) <latexit 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y = +1 <latexit 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y = 1 <latexit 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=) <latexit 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=) <latexit 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VAE, GAN, di↵usion <latexit 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k-means, DBScan <latexit 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PCA, t-SNE, UMAP <latexit 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Self-supervised learning: Demain, dès l'aube, à l'heure où blanchit la campagne, Je partirai. Vois- tu, je sais que tu m’attends. J'irai par la forêt, j'irai par la montagne. Je ne puis demeurer loin de toi plus longtemps. Je marcherai les yeux fixés sur mes pensées,… Demain, dès l'aube, à l'heure où blanchit la campagne, Je partirai. Vois- tu, je sais que tu m’attends. J'irai par la forêt, j'irai par la montagne. Je ne puis demeurer loin de toi plus longtemps. Je marcherai les yeux fixés sur mes pensées,… <latexit 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? ? ? <latexit 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? ? ? <latexit 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=)
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xi 2 Rd <latexit 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yi 2 R <latexit 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y ⇡ f✓(x) <latexit 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Empirical risk minimization: <latexit 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min ✓ 1 n n X i=1 `(f✓(xi), yi) <latexit 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f✓ <latexit 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f✓0 <latexit 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y <latexit 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Regression: yi 2 R <latexit 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Least square: <latexit sha1_base64="GTs/6CYLabKgJFBv6ol/ZD+zFtI=">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</latexit> `(y, y0) = (y y0)2
b, log(a), p a . . . ) and their derivatives cost O(1). <latexit sha1_base64="XBU0O16nV93z/lhQgn5HZRGW7gQ=">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</latexit> Setup: E : Rd ! R computable in K operations. <latexit sha1_base64="BLbI91EqNObAlbI/2RJe4xs+gGo=">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</latexit> Question: What is the complexity of computing rE : Rd ! Rd?
b, log(a), p a . . . ) and their derivatives cost O(1). <latexit sha1_base64="XBU0O16nV93z/lhQgn5HZRGW7gQ=">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</latexit> Setup: E : Rd ! R computable in K operations. <latexit sha1_base64="BLbI91EqNObAlbI/2RJe4xs+gGo=">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</latexit> Question: What is the complexity of computing rE : Rd ! Rd? Finite di↵erences: <latexit sha1_base64="IJjbmZX1RxHFJ6LjH+Uz9oKFiHc=">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</latexit> rE(✓) ⇡ 1 " (E(✓ + " 1) E(✓), . . . E(✓ + " d) E(✓)) <latexit sha1_base64="elfmtYFgpa8JKGP5IFiZSMc3eME=">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</latexit> K(d + 1) operations, intractable for large d.
b, log(a), p a . . . ) and their derivatives cost O(1). Seppo Linnainmaa This algorithm is reverse mode automatic di↵erentiation [Seppo Linnainmaa, 1970] Theorem: there is an algorithm to compute rE in O(K) operations. <latexit sha1_base64="XBU0O16nV93z/lhQgn5HZRGW7gQ=">AAA9x3ictVttc9u4EUaub5f0Ldd+7EyHrZ3OXSf12G6m7dyNZy6xncQXX+JEspO7U5LRCy0zkUSFpGwnOn/oT+ov6M/o9Fv7rf3Uv9DdBUCAEsgF3NQc2yCE59nFEljsAlRvOkryYn3971c++M53v/f9H3x49doPf/Tjn/z0+kc/O8rTWdaPD/vpKM2e9bp5PEom8WGRFKP42TSLu+PeKH7ae72Nnz89jbM8SSft4u00fj7uDifJcdLvFlD18vqDThGfF73jeSsuZtNPL6LVzm4cfRp1njx5MYg6WTI8KbpZlp5hzWrUT8fTWdEF9iiZRKsPVqN0GmfEla+9vL6yvrZOP9FyYUMVVoT6OUg/+uWm6IiBSEVfzMRYxGIiCiiPRFfkcH0jNsS6mELdczGHugxKCX0eiwtxDbAzaBVDiy7Uvoa/Q7j7RtVO4B45c0L3QcoIfjNARuIGYFJol0EZpUX0+YyYsbaOe06cqNtb+N9TXGOoLcQJ1HI43dIXh30pxLH4E/UhgT5NqQZ711csM7IKah5ZvSqAYQp1WB7A5xmU+4TUdo4Ik1Pf0bZd+vxf1BJr8b6v2s7Ev0nLG3BFoqV6n5YMXXFK/BE9zRl8JvUZgeQhMMSqj1g6I1uPqfcTaD+H+odwXVBJ26QH15xqLxqR23C5kNss8h5cLuQ9FrkPlwu5zyIP4HIhDxQSsRnZ3I1vweXCt1jJj+FyIR+zyCdwuZBPWOQRXC7kEYv8Gi4X8msWeRcuF/Iui3wAlwv5gEW24XIh2yzyEC4X8pBF7sLlQu4qZP1MzeBKiSdhZuVtKFdloKcYQc1tVr875B1d2Dsec7pfg+Vn9Q78d2N3PGwa12B3PcbdcQ2WH3n3wEe6sbwvuk+riQt7n8XuwQhwY/dY7BfiVQ32C4+Z9roGy8+1fWjnxvLe90u4c2O/ZLEPoeTG8mvUI6hxYx95rBjTGuwBi30s3tRgfbx+VoPl/X4L/Ioby69TbWjvxvp401kNlvenRxDBuLH8avUUat3Ypyz2mTivwT5jsV+Bd3djv/JYYd/VYPUae41WkCHFIzHM2Ca2bjkrsTQFti4jf1SuLSOKjXtQz2GGJWZImDGLuFci7nki9kvEvrdeeelHc4p3eSmtEtHyRPTKtQlLBdt+ULbH0sgDsVMidhYQTREpPmvdl1OKLnQNhyzKlQtLPn1KS/+NpViNh2bPqxGPKgg5tk9o5N+kbAkzKLRUE9tJucZLZET3TYgzyt50L7UMHleUXsFGnbOongPVY1FvHai3LGrmQM1Y1KkDdcqizMy3cR2PEWDsj89iTndyBMgYuf6KICq4DavOfZijEYyfA4gCn1DNI/jfotybu5o0w2we10nc5Xhe8cQZlOZiBepNVrhD+fWIZlgMmsmWj1SOj3e4tzFXc0564YtyJY/KHRN/noT0GZY8GC1GNJ/CeB5QzQVFd7IUhr9fzntdCsPvksUvKIqXpTB8obQvLqF7W2Hbl8C2YDZNlfVNOZRD7r9IDl2+Rqsuelx8qmM1ZpDvPJB/Tz2ZvUs8l20qSfuYchhHbvUvr/QvhMPYObfsHMaC0ZOMenUpCu7JROW9phyqQ0qr6ETpYe5Cnwy2Gagno8thHAcQcW1Tzj23yqGjd1r2xpTDOI6E3Pe8oEhel8M4hnQv7WHKYRy429JVeb4ph3p2tIDMnU051KtPaBcY94DkmJc1JirKKE6aKbaE4oPm3Ro75l9ex3DP5kWZIzQzmdi2nqdXrmXNGul4IQavVgTqgfHFzIrBqhxzscnmV1KHorK+L/OYNR4tvw9WjGD2yzMAbs98BBrqPQn03iNg3GCzrmrPNG6TxeEoOV5AdVRtwUaLRq7cNarWvaRaLi8zvTV27JC/zmnsTSkm3CfLcnbYr33CdYychfYrFuL5Qmz3Ts3XqvXXWdx0ATEtR1qfToTkSVpznuqyesuy8Q11ylPAJc98zPjF3eZj5W0w50nJF6EuTTLtdnofya7DdfWmMHvc8rOInij6q1PyGgmdSOVsFqp3i2U0Pqd7w31IZ3IoQ3L04TlGimUq5KkZ7qLjfnpEHtX2t5xstJfeoZPlnLyu9sfN6KGFHjrQ4TnONqwYD6HUhpzhEO7aHlnOtdJWKVk8E78rT0dTeoLNGf2o4iE1h/Q3ccVDNmXZJxWWM0DjaJBZuj/HIo/Gd5aY+KzfpY/JXaue/wad3Orz7S6N8frRXL8TMyCpmyQ1olkjT3Xl3aIEqcHc+ckmxa/NvUR5IRLRh3JSX1iSpV0mdOIfUwY7pch4RLONmx3V1vb+1OInWtKB0GfneJqdkoeMyP9FsD6lNCYj+rXfHdAn6NIjjMhH+vidpIxuXLFOwo4xE8clQr7VYMZbTL5sRvI1rz27chqLMmOQ68DFwtjWNtmnWDAmqZny7mZuN68+iDTvSdijRDKasfIxyf+E/upfPU5WlkYEWhifQK58net5pJSzoI26tMo3+yDd1tZytdThhdLarH9Gp9WKZjuUcaE+uFoPQHKf7qUsHCUZ6Z0vtZHraNNuLjJPF+yIvT2mLF76/aFagVHvm7RKrtCc69AoGcIoKMosQrfldpEX5TbLqrL7cef/F3Zj66rVkDESZgdXWojb348pW7O1HMGoluP3Nc0mt9WzhVbNciY0FsfWXP4Wan8Ff7Xe+t6Pp1fxCndoDEgGc2csImuipRZ+su5UZOmRqbnMvZFnxqRuZddcJr+W3s3k2KfBLAc0as7VroUuX4bjlcXxytOGbTprNFbU9doTvWRzi7Y6rfSVFyKtHcA8Y5n5iEyjEg8t7VzKj3XAsvI5vka9Y7nWWa4uzFb7NMCe8z5I91xfnN3flqt7JO5SbNOnCEzmLwOapQnFXLq2OVOTDCj5lvKv9uzvUA1K75EHRWb5HifOGHnq1KfrotT0N2plS8nPG4+g31s6U220j+1Q+fdLyDHNiZzmpUbcohax0t/WI1rwSGtWzBHRzn+XYioZdzTnzHZr80yiSjxh8k05q4wsmSlMyP7cztveUva6Z+WvEeWEMxVd94Ar/Akjg8TonQR3ZJnTE8JVTp4kyIi2R/5z2U/JU7yJpdEaaT0XWx4+Rma9ZqzbY0v3WPftt9ASrW6euqsFL2/kLZGTd5kTvS6tamMVo84X7i/H1VWrXPW+yQ6zBbnGHjNqY2cWJsurYjriM28pUqMwKRLjIyWsFyH6h2keorM8nfJl1q01c3WnQfqYE8qXuPdAEeGK7j52RnOfMP3oLfH1CGuzyRqOCXfjUrU/YHta3JW6urQOydqrjavRyFqJ6lYKzW6vFsZ/Sw8Zk/cbCW7PRra2de9UshR+F0Yy9IV8o7cuP7Q5P4ML/0bClR1qiT57hy2Ib2+LbbH7Ht6GeKPKckczohr0BYOF3Lur+llt0WyjNxa7ze8jwV9GArbmtE9oJQ3VXTLzmtvs/vxn5AUyEbPam5bhfbCl8D1ZlhTSn4Q8G9+bROjv4oT2RUvw6UlVir8cea7B9eJY6O80hfVBs/M9qEoIkaHfY/B75qZ1uCxbUrO9lqX4ypCrgD5x0Tg8+avPVUw7Hw+VWU/k/UtA73DcwK5Xi/+1H1qOkRQuy1daTt81e+Xx1GW7WO3IYjwcPmeMNJ/RXC/RX2Za9s5ES255Mu6Lgp5UavXm/fNjPGrGgJY1F3IflNdO4u1RZPT1ZcFzAZcOqfiP+OsV/tsIb0qOOj1CmPQ5RT2bbsGz6W9cunqnP/PRyfDU6VRlM3lEi96I3RZ74i78bpcRYOjbofK7lPI/Yt3fnx1A7TF5D72LLncOOlQX0+6HOUUb0L3aY3x5fWVj8VvIy4WjzbWNP6zdery58vkd9Q3lD8UvxK8hL9kQfxSfi/vQ30PQ6S/ib+If4p9be1vp1unWuWz6wRWF+bmo/Gz9+b9va/Dv</latexit> Setup: E : Rd ! R computable in K operations. <latexit sha1_base64="BLbI91EqNObAlbI/2RJe4xs+gGo=">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</latexit> Question: What is the complexity of computing rE : Rd ! Rd? Finite di↵erences: <latexit sha1_base64="IJjbmZX1RxHFJ6LjH+Uz9oKFiHc=">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</latexit> rE(✓) ⇡ 1 " (E(✓ + " 1) E(✓), . . . E(✓ + " d) E(✓)) <latexit sha1_base64="elfmtYFgpa8JKGP5IFiZSMc3eME=">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</latexit> K(d + 1) operations, intractable for large d.
= P k xk✓k x f(x) = 0 x y y = f(x) <latexit sha1_base64="+XMTGU2h84i8kefw5ntE/6bvDrk=">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</latexit> y = hx, ✓i <latexit 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✓d <latexit 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✓1 <latexit 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. . . <latexit 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Regression <latexit 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Classification: <latexit 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min ✓ , 1 n n X i=1 `(hxi, ✓i i, yi) <latexit 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Convex optimization:
= P k xk✓k <latexit 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Deep learning methods: learn '(x)! <latexit 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Kernel methods: replace x by '(x) 2 RD <latexit 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(D d, even D = 1!) x f(x) = 0 x y y = f(x) <latexit 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y = hx, ✓i <latexit 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✓d <latexit 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✓1 <latexit 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. . . <latexit 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Regression <latexit 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Classification: <latexit 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min ✓ , 1 n n X i=1 `(hxi, ✓i i, yi) <latexit sha1_base64="PvseIPRi4Nf4IH/9H8AceNGnDTM=">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</latexit> Convex optimization:
sha1_base64="4q707XOseaJG4SEM7kRUBlKiggY=">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</latexit> w1 <latexit sha1_base64="dTEPGRjwpObKto22Pdj89jMy0R4=">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</latexit> a1 p = 6 neurons p = 30 neurons p = 100 neurons <latexit sha1_base64="v5o5dJsoyPefg3wihB9aJpDhTXc=">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</latexit> Input y = f(x) <latexit sha1_base64="ttgglA5lmQEZUY9EO2WF1erSv+8=">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</latexit> ! sum of “ridge” functions (hx, wi + b) <latexit 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f✓(x) , p X s=1 as (hx, ws i + bs) <latexit 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w1 <latexit sha1_base64="dTEPGRjwpObKto22Pdj89jMy0R4=">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</latexit> a1 p = 6 neurons p = 30 neurons p = 100 neurons <latexit sha1_base64="v5o5dJsoyPefg3wihB9aJpDhTXc=">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</latexit> Input y = f(x) <latexit sha1_base64="ttgglA5lmQEZUY9EO2WF1erSv+8=">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</latexit> ! sum of “ridge” functions (hx, wi + b) <latexit 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f✓(x) , p X s=1 as (hx, ws i + bs) <latexit 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wp <latexit 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ap
If f is continuous on a compact ⌦, for all " > 0 <latexit 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! non quantitative . . . no free lunch. <latexit sha1_base64="Jz0OwNMv5k//RDHXco0G6196vK0=">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</latexit> for p large enough, there exists ✓ such that <latexit sha1_base64="xfYOcK/7SLaT3sOdjnYPuyrhJIM=">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</latexit> 8 x 2 ⌦, |f✓(x) f(x)| 6 "
<latexit 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If f is continuous on a compact ⌦, for all " > 0 <latexit sha1_base64="uvCA28W4YytRhUCzr+wZy5dsPHE=">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</latexit> ! non quantitative . . . no free lunch. <latexit sha1_base64="Jz0OwNMv5k//RDHXco0G6196vK0=">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</latexit> for p large enough, there exists ✓ such that <latexit sha1_base64="xfYOcK/7SLaT3sOdjnYPuyrhJIM=">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</latexit> 8 x 2 ⌦, |f✓(x) f(x)| 6 " Barron’s functions: <latexit 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||f||B , R Rd ||!||| ˆ f(!)|d! < +1 <latexit 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||f f✓ ||L2(⌦) 6 2diam(⌦)||f||B p p <latexit sha1_base64="hO/zv54wxyh8DCE0R1zGL/84DG0=">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</latexit> Theorem: for p large, there exists ✓ such that
<latexit 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If f is continuous on a compact ⌦, for all " > 0 <latexit 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! non quantitative . . . no free lunch. <latexit 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! non-constructive. <latexit 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||f f✓ || <latexit sha1_base64="Jz0OwNMv5k//RDHXco0G6196vK0=">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</latexit> for p large enough, there exists ✓ such that <latexit sha1_base64="xfYOcK/7SLaT3sOdjnYPuyrhJIM=">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</latexit> 8 x 2 ⌦, |f✓(x) f(x)| 6 " Barron’s functions: <latexit 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||f||B , R Rd ||!||| ˆ f(!)|d! < +1 <latexit 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||f f✓ ||L2(⌦) 6 2diam(⌦)||f||B p p <latexit sha1_base64="hO/zv54wxyh8DCE0R1zGL/84DG0=">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</latexit> Theorem: for p large, there exists ✓ such that <latexit sha1_base64="h3UjZpHYqKKTnXvhALF0qypbZ8I=">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</latexit> ! for p “large enough” <latexit 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gradient descent works <latexit 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[Chizat-Bach 2018] <latexit 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f✓ <latexit 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f
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Positional encoding + <latexit 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Points cloud <latexit 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x1 <latexit 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x2 Demain, dès l'aube, à l'heure où blanchit la campagne, Je partirai. Vois- tu, je sais que tu m’attends. J'irai par la forêt, j'irai par la montagne. Je ne puis demeurer loin de toi plus longtemps. Je marcherai les yeux fixés sur mes pensées,… <latexit sha1_base64="Fgw+vWgPriclgLxpVoHDEicBDLw=">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</latexit> {xi }i
<latexit 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Positional encoding + <latexit 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Points cloud <latexit 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x1 <latexit 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x2 Demain, dès l'aube, à l'heure où blanchit la campagne, Je partirai. Vois- tu, je sais que tu m’attends. J'irai par la forêt, j'irai par la montagne. Je ne puis demeurer loin de toi plus longtemps. Je marcherai les yeux fixés sur mes pensées,… <latexit 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{xi }i <latexit sha1_base64="TxVb5GdPgjAMGN+p3Ec+OBLQ8cg=">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</latexit> ˜ xi = xi + P j eKxi,Qxj V xj P j eKxi,Qxj <latexit 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xi <latexit 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Residual <latexit sha1_base64="kiilw7sVPY4P3AU1/BWI0LZRksc=">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</latexit> Replace convolution by attention: