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• W2752988442 abstract "We consider the problem of computing a restricted nonnegative matrix factorization (NMF) of an mtimes n matrix X. Specifically, we seek a factorization Xapprox BC, where the k columns of B are a subset of those from X and CinRe_{geq 0}^{ktimes n}. Equivalently, given the matrix X, consider the problem of finding a small subset, S, of the columns of X such that the conic hull of S eps-approximates the conic hull of the columns of X, i.e., the distance of every column of X to the conic hull of the columns of S should be at most an eps-fraction of the angular diameter of X. If k is the size of the smallest eps-approximation, then we produce an O(k/eps^{2/3}) sized O(eps^{1/3})-approximation, yielding the first provable, polynomial time eps-approximation for this class of NMF problems, where also desirably the approximation is independent of n and m. Furthermore, we prove an approximate conic Caratheodory theorem, a general sparsity result, that shows that any column of X can be eps-approximated with an O(1/eps^2) sparse combination from S. Our results are facilitated by a reduction to the problem of approximating convex hulls, and we prove that both the convex and conic hull variants are d-sum-hard, resolving an open problem. Finally, we provide experimental results for the convex and conic algorithms on a variety of feature selection tasks." @default.
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• W2752988442 date "2017-12-01" @default.
• W2752988442 modified "2023-09-24" @default.
• W2752988442 title "Sparse Approximate Conic Hulls" @default.
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