Scalable Methods for Nonnegative Matrix Factorizations of Near-separable Tall-and-skinny Matrices
نویسندگان
چکیده
Introduction to (near-separable) NMF • NMF Problem: X ∈ Rm×n + is a matrix with nonnegative entries, and we want to compute a nonnegative matrix factorization (NMF) X = WH, where W ∈ Rm×r + and H ∈ Rr×n + . When r < m, this problem is NP-hard. • A separable matrix is one that admits a nonnegative factorization where W = X(:,K), i.e. W is just consists of some subset of the columns of X . A near-separable matrix is one where X = X(:,K)H+N , where N represents noise. The set K of columns are called extreme columns. • Under the near-separable assumptions, there are efficient algorithms for computing the NMF. The algorithms typically proceed as follows: 1. Determine the extreme columns, indexed by K, and let W = X(:,K). 2. With W fixed, solve H = arg minY ∈Rr×n + ‖X −WY ‖.
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