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Matrices and tensors, or multidimensional arrays, are highly versatile data structures. They can store a variety of formats: image, video, table, and sensing data. By decomposing such matrices and tensors, we can extract insight from the data: patterns, features, etc. In this lecture, I introduce singular value decomposition (SVD) for matrices, non-negative matrix factorization (NMF) for non-negative matrices, and CP and Tucker decomposition for tensors. The objective is not only to understand the various decomposition methods in a piecemeal manner but also to acquire the know-how to select the appropriate decomposition method according to the situation and various constraints in real-world scenarios while focusing on the properties of each decomposition. For downstream applications of the methods, the lecture will cover subspace methods (CLAFIC) for classification tasks and EM-based methods for determining missing values in given input data. In the last segment of the lecture, I will cover some of the difficulties of tensor decomposition (TD). TD suffers from difficulties that do not occur in SVD, such as ill-posedness and NP-hardness in optimization, and will discuss recent research trends for avoiding these difficulties.

July 26, 2024