5,668 research outputs found
Intersecting Faces: Non-negative Matrix Factorization With New Guarantees
Non-negative matrix factorization (NMF) is a natural model of admixture and
is widely used in science and engineering. A plethora of algorithms have been
developed to tackle NMF, but due to the non-convex nature of the problem, there
is little guarantee on how well these methods work. Recently a surge of
research have focused on a very restricted class of NMFs, called separable NMF,
where provably correct algorithms have been developed. In this paper, we
propose the notion of subset-separable NMF, which substantially generalizes the
property of separability. We show that subset-separability is a natural
necessary condition for the factorization to be unique or to have minimum
volume. We developed the Face-Intersect algorithm which provably and
efficiently solves subset-separable NMF under natural conditions, and we prove
that our algorithm is robust to small noise. We explored the performance of
Face-Intersect on simulations and discuss settings where it empirically
outperformed the state-of-art methods. Our work is a step towards finding
provably correct algorithms that solve large classes of NMF problems
Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix
The positive semidefinite rank of a nonnegative -matrix~ is
the minimum number~ such that there exist positive semidefinite -matrices , such that S(k,\ell) =
\mbox{tr}(A_k^* B_\ell).
The most important, lower bound technique for nonnegative rank is solely
based on the support of the matrix S, i.e., its zero/non-zero pattern. In this
paper, we characterize the power of lower bounds on positive semidefinite rank
based on solely on the support.Comment: 9 page
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