5,721 research outputs found
Using Underapproximations for Sparse Nonnegative Matrix Factorization
Nonnegative Matrix Factorization consists in (approximately) factorizing a
nonnegative data matrix by the product of two low-rank nonnegative matrices. It
has been successfully applied as a data analysis technique in numerous domains,
e.g., text mining, image processing, microarray data analysis, collaborative
filtering, etc.
We introduce a novel approach to solve NMF problems, based on the use of an
underapproximation technique, and show its effectiveness to obtain sparse
solutions. This approach, based on Lagrangian relaxation, allows the resolution
of NMF problems in a recursive fashion. We also prove that the
underapproximation problem is NP-hard for any fixed factorization rank, using a
reduction of the maximum edge biclique problem in bipartite graphs.
We test two variants of our underapproximation approach on several standard
image datasets and show that they provide sparse part-based representations
with low reconstruction error. Our results are comparable and sometimes
superior to those obtained by two standard Sparse Nonnegative Matrix
Factorization techniques.Comment: Version 2 removed the section about convex reformulations, which was
not central to the development of our main results; added material to the
introduction; added a review of previous related work (section 2.3);
completely rewritten the last part (section 4) to provide extensive numerical
results supporting our claims. Accepted in J. of Pattern Recognitio
Conditional Gradient Algorithms for Rank-One Matrix Approximations with a Sparsity Constraint
The sparsity constrained rank-one matrix approximation problem is a difficult
mathematical optimization problem which arises in a wide array of useful
applications in engineering, machine learning and statistics, and the design of
algorithms for this problem has attracted intensive research activities. We
introduce an algorithmic framework, called ConGradU, that unifies a variety of
seemingly different algorithms that have been derived from disparate
approaches, and allows for deriving new schemes. Building on the old and
well-known conditional gradient algorithm, ConGradU is a simplified version
with unit step size and yields a generic algorithm which either is given by an
analytic formula or requires a very low computational complexity. Mathematical
properties are systematically developed and numerical experiments are given.Comment: Minor changes. Final version. To appear in SIAM Revie
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