11,266 research outputs found
An Almost Optimal Algorithm for Computing Nonnegative Rank
Here, we give an algorithm for deciding if the nonnegative rank of a matrix M of dimension m \times n$ is at most r which runs in time (nm)[superscript O(r2)]. This is the first exact algorithm that runs in time singly exponential in r. This algorithm (and earlier algorithms) are built on methods for finding a solution to a system of polynomial inequalities (if one exists). Notably, the best algorithms for this task run in time exponential in the number of variables but polynomial in all of the other parameters (the number of inequalities and the maximum degree). Hence, these algorithms motivate natural algebraic questions whose solution have immediate algorithmic implications: How many variables do we need to represent the decision problem, and does M have nonnegative rank at most r? A naive formulation uses nr + mr variables and yields an algorithm that is exponential in n and m even for constant r. Arora et al. [Proceedings of STOC, 2012, pp. 145--162] recently reduced the number of variables to 2r[superscript 2] 2[superscript r], and here we exponentially reduce the number of variables to 2r[superscript 2] and this yields our main algorithm. In fact, the algorithm that we obtain is nearly optimal (under the exponential time hypothesis) since an algorithm that runs in time (nm)[superscript o(r)] would yield a subexponential algorithm for 3-SAT [Proceedings of STOC, 2012, pp. 145--162]. Our main result is based on establishing a normal form for nonnegative matrix factorization---which in turn allows us to exploit algebraic dependence among a large collection of linear transformations with variable entries. Additionally, we also demonstrate that nonnegative rank cannot be certified by even a very large submatrix of M, and this property also follows from the intuition gained from viewing nonnegative rank through the lens of systems of polynomial inequalities.National Science Foundation (U.S.) (Computing and Innovation Fellowship)National Science Foundation (U.S.) (grant DMS-0835373
Computing approximate PSD factorizations
We give an algorithm for computing approximate PSD factorizations of
nonnegative matrices. The running time of the algorithm is polynomial in the
dimensions of the input matrix, but exponential in the PSD rank and the
approximation error. The main ingredient is an exact factorization algorithm
when the rows and columns of the factors are constrained to lie in a general
polyhedron. This strictly generalizes nonnegative matrix factorizations which
can be captured by letting this polyhedron to be the nonnegative orthant.Comment: 10 page
Generalized Separable Nonnegative Matrix Factorization
Nonnegative matrix factorization (NMF) is a linear dimensionality technique
for nonnegative data with applications such as image analysis, text mining,
audio source separation and hyperspectral unmixing. Given a data matrix and
a factorization rank , NMF looks for a nonnegative matrix with
columns and a nonnegative matrix with rows such that .
NMF is NP-hard to solve in general. However, it can be computed efficiently
under the separability assumption which requires that the basis vectors appear
as data points, that is, that there exists an index set such that
. In this paper, we generalize the separability
assumption: We only require that for each rank-one factor for
, either for some or for
some . We refer to the corresponding problem as generalized separable NMF
(GS-NMF). We discuss some properties of GS-NMF and propose a convex
optimization model which we solve using a fast gradient method. We also propose
a heuristic algorithm inspired by the successive projection algorithm. To
verify the effectiveness of our methods, we compare them with several
state-of-the-art separable NMF algorithms on synthetic, document and image data
sets.Comment: 31 pages, 12 figures, 4 tables. We have added discussions about the
identifiability of the model, we have modified the first synthetic
experiment, we have clarified some aspects of the contributio
Factoring nonnegative matrices with linear programs
This paper describes a new approach, based on linear programming, for
computing nonnegative matrix factorizations (NMFs). The key idea is a
data-driven model for the factorization where the most salient features in the
data are used to express the remaining features. More precisely, given a data
matrix X, the algorithm identifies a matrix C such that X approximately equals
CX and some linear constraints. The constraints are chosen to ensure that the
matrix C selects features; these features can then be used to find a low-rank
NMF of X. A theoretical analysis demonstrates that this approach has guarantees
similar to those of the recent NMF algorithm of Arora et al. (2012). In
contrast with this earlier work, the proposed method extends to more general
noise models and leads to efficient, scalable algorithms. Experiments with
synthetic and real datasets provide evidence that the new approach is also
superior in practice. An optimized C++ implementation can factor a
multigigabyte matrix in a matter of minutes.Comment: 17 pages, 10 figures. Modified theorem statement for robust recovery
conditions. Revised proof techniques to make arguments more elementary.
Results on robustness when rows are duplicated have been superseded by
arxiv.org/1211.668
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