1,346 research outputs found

    Lifts of convex sets and cone factorizations

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    In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or 'lift' of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual. This generalizes a theorem of Yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts of convex sets can also be characterized similarly. When the cones live in a family, our results lead to the definition of the rank of a convex set with respect to this family. We present results about this rank in the context of cones of positive semidefinite matrices. Our methods provide new tools for understanding cone lifts of convex sets.Comment: 20 pages, 2 figure

    Computing a Nonnegative Matrix Factorization -- Provably

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    In the Nonnegative Matrix Factorization (NMF) problem we are given an n×mn \times m nonnegative matrix MM and an integer r>0r > 0. Our goal is to express MM as AWA W where AA and WW are nonnegative matrices of size n×rn \times r and r×mr \times m respectively. In some applications, it makes sense to ask instead for the product AWAW to approximate MM -- i.e. (approximately) minimize \norm{M - AW}_F where \norm{}_F denotes the Frobenius norm; we refer to this as Approximate NMF. This problem has a rich history spanning quantum mechanics, probability theory, data analysis, polyhedral combinatorics, communication complexity, demography, chemometrics, etc. In the past decade NMF has become enormously popular in machine learning, where AA and WW are computed using a variety of local search heuristics. Vavasis proved that this problem is NP-complete. We initiate a study of when this problem is solvable in polynomial time: 1. We give a polynomial-time algorithm for exact and approximate NMF for every constant rr. Indeed NMF is most interesting in applications precisely when rr is small. 2. We complement this with a hardness result, that if exact NMF can be solved in time (nm)o(r)(nm)^{o(r)}, 3-SAT has a sub-exponential time algorithm. This rules out substantial improvements to the above algorithm. 3. We give an algorithm that runs in time polynomial in nn, mm and rr under the separablity condition identified by Donoho and Stodden in 2003. The algorithm may be practical since it is simple and noise tolerant (under benign assumptions). Separability is believed to hold in many practical settings. To the best of our knowledge, this last result is the first example of a polynomial-time algorithm that provably works under a non-trivial condition on the input and we believe that this will be an interesting and important direction for future work.Comment: 29 pages, 3 figure

    Support-based lower bounds for the positive semidefinite rank of a nonnegative matrix

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    The positive semidefinite rank of a nonnegative (m×n)(m\times n)-matrix~SS is the minimum number~qq such that there exist positive semidefinite (q×q)(q\times q)-matrices A1,…,AmA_1,\dots,A_m, B1,…,BnB_1,\dots,B_n 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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