Sparse Approximate Conic Hulls

We consider the problem of computing a restricted nonnegative matrix factorization (NMF) of an m x n matrix X. Specifically, we seek a factorization X approximate to BC, where the k columns of B are a subset of those from X and C is an element of R(>= 0)(kxn)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 epsilon-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 epsilon-fraction of the angular diameter of X. If k is the size of the smallest epsilon-approximation, then we produce an O(k/epsilon(2/3)) sized O(epsilon(1/3))-approximation, yielding the first provable, polynomial time epsilon-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 epsilon-approximated with an O(1/epsilon(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.