
Sparse Approximate Conic Hulls 
Author(s): 
Van Buskirk, Gregory 

Raichel, Benjamin 

201712 

Thesis 
Keywords:  Nonnegative matrices Factorization (Mathematics) Approximation algorithms Conic sections 

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 ≈ BC, where the k columns of B are a subset of those from X and C ∈ Re_{≥ 0}^{k x 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 εapproximates 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 εfraction of the angular diameter of X. If k is the size of the smallest εapproximation, then we produce an O(k/ε^{2/3}) sized O(ε^{1/3})approximation, yielding the first provable, polynomial time εapproximation for this class of NMF problems, where also desirably the approximation is independent of n and m. Furthermore, we prove an approximate conic Carathéodory theorem, a general sparsity result, that shows that any column of X can be εapproximated with an O(1/ε^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 dSUMhard, resolving an open problem. Finally, we provide experimental results for the convex and conic algorithms on a variety of feature selection tasks. 

MSCS 

Masters 

http://hdl.handle.net/10735.1/5688 

Copyright ©2017 is held by the author. Digital access to this material is made possible by the Eugene McDermott Library. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. 

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Computer Science 
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