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arXiv · 1401.6024

Matrix factorization with Binary Components

Martin Slawski·Matthias Hein·Pavlo Lutsik
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Abstract

Motivated by an application in computational biology, we consider low-rank matrix factorization with $\{0,1\}$-constraints on one of the factors and optionally convex constraints on the second one. In addition to the non-convexity shared with other matrix factorization schemes, our problem is further complicated by a combinatorial constraint set of size $2^{m \cdot r}$, where $m$ is the dimension of the data points and $r$ the rank of the factorization. Despite apparent intractability, we provide - in the line of recent work on non-negative matrix factorization by Arora et al. (2012) - an algorithm that provably recovers the underlying factorization in the exact case with $O(m r 2^r + mnr + r^2 n)$ operations for $n$ datapoints. To obtain this result, we use theory around the Littlewood-Offord lemma from combinatorics.

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Martin Slawski, Matthias Hein, Pavlo Lutsik. 2014-01-23. Matrix factorization with Binary Components. https://arxiv.org/abs/1401.6024

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