Bounded Simplex-Structured Matrix Factorization
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In this paper, we propose a new low-rank matrix factorization model dubbed bounded simplex-structured matrix factorization (BSSMF). Given an input matrix X and a factorization rank r, BSSMF looks for a matrix W with r columns and a matrix H with r rows such that X ≈ WH where the entries in each column of W are bounded, that is, they belong to given intervals, and the columns of H belong to the probability simplex, that is, H is column stochastic. BSSMF generalizes nonnegative matrix factorization (NMF), and simplex-structured matrix factorization (SSMF). BSSMF is particularly well suited when the entries of the input matrix X belong to a given interval; for example when the rows of X represent images, or X is a rating matrix such as in the Netflix and MovieLens data sets where the entries of X belong to the interval [1,5]. The simplex-structured matrix H not only leads to an easily understandable decomposition providing a soft clustering of the columns of X, but implies that the entries of each column of WH belong to the same intervals as the columns of W. In this paper, we first propose a fast algorithm for BSSMF, even in the presence of missing data in X. Then we provide identifiability conditions for BSSMF, that is, we provide conditions under which BSSMF admits a unique decomposition, up to trivial ambiguities. Finally, we illustrate the effectiveness of BSSMF on two applications: extraction of features in a set of images, and the matrix completion problem for recommender systems.
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