Heuristics for Exact Nonnegative Matrix Factorization

The exact nonnegative matrix factorization (exact NMF) problem is the following: given an $m$-by-$n$ nonnegative matrix $X$ and a factorization rank $r$, find, if possible, an $m$-by-$r$ nonnegative matrix $W$ and an $r$-by-$n$ nonnegative matrix $H$ such that $X = WH$. In this paper, we propose two heuristics for exact NMF, one inspired from simulated annealing and the other from the greedy randomized adaptive search procedure. We show that these two heuristics are able to compute exact nonnegative factorizations for several classes of nonnegative matrices (namely, linear Euclidean distance matrices, slack matrices, unique-disjointness matrices, and randomly generated matrices) and as such demonstrate their superiority over standard multi-start strategies. We also consider a hybridization between these two heuristics that allows us to combine the advantages of both methods. Finally, we discuss the use of these heuristics to gain insight on the behavior of the nonnegative rank, i.e., the minimum factorization rank such that an exact NMF exists. In particular, we disprove a conjecture on the nonnegative rank of a Kronecker product, propose a new upper bound on the extension complexity of generic $n$-gons and conjecture the exact value of (i) the extension complexity of regular $n$-gons and (ii) the nonnegative rank of a submatrix of the slack matrix of the correlation polytope.Comment: 32 pages, 2 figures, 16 table

Computing a Nonnegative Matrix Factorization -- Provably

In the Nonnegative Matrix Factorization (NMF) problem we are given an $n \times m$ nonnegative matrix $M$ and an integer $r > 0$. Our goal is to express $M$ as $A W$ where $A$ and $W$ are nonnegative matrices of size $n \times r$ and $r \times m$ respectively. In some applications, it makes sense to ask instead for the product $AW$ to approximate $M$ -- 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 $A$ and $W$ 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 $r$. Indeed NMF is most interesting in applications precisely when $r$ is small. 2. We complement this with a hardness result, that if exact NMF can be solved in time $(nm)^{o(r)}$, 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 $n$, $m$ and $r$ 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

Learning Topic Models - Going beyond SVD

Topic Modeling is an approach used for automatic comprehension and classification of data in a variety of settings, and perhaps the canonical application is in uncovering thematic structure in a corpus of documents. A number of foundational works both in machine learning and in theory have suggested a probabilistic model for documents, whereby documents arise as a convex combination of (i.e. distribution on) a small number of topic vectors, each topic vector being a distribution on words (i.e. a vector of word-frequencies). Similar models have since been used in a variety of application areas; the Latent Dirichlet Allocation or LDA model of Blei et al. is especially popular. Theoretical studies of topic modeling focus on learning the model's parameters assuming the data is actually generated from it. Existing approaches for the most part rely on Singular Value Decomposition(SVD), and consequently have one of two limitations: these works need to either assume that each document contains only one topic, or else can only recover the span of the topic vectors instead of the topic vectors themselves. This paper formally justifies Nonnegative Matrix Factorization(NMF) as a main tool in this context, which is an analog of SVD where all vectors are nonnegative. Using this tool we give the first polynomial-time algorithm for learning topic models without the above two limitations. The algorithm uses a fairly mild assumption about the underlying topic matrix called separability, which is usually found to hold in real-life data. A compelling feature of our algorithm is that it generalizes to models that incorporate topic-topic correlations, such as the Correlated Topic Model and the Pachinko Allocation Model. We hope that this paper will motivate further theoretical results that use NMF as a replacement for SVD - just as NMF has come to replace SVD in many applications

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

A geometric approach to archetypal analysis and non-negative matrix factorization

Archetypal analysis and non-negative matrix factorization (NMF) are staples in a statisticians toolbox for dimension reduction and exploratory data analysis. We describe a geometric approach to both NMF and archetypal analysis by interpreting both problems as finding extreme points of the data cloud. We also develop and analyze an efficient approach to finding extreme points in high dimensions. For modern massive datasets that are too large to fit on a single machine and must be stored in a distributed setting, our approach makes only a small number of passes over the data. In fact, it is possible to obtain the NMF or perform archetypal analysis with just two passes over the data.Comment: 36 pages, 13 figure

Algorithms for Positive Semidefinite Factorization

This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an $m$-by-$n$ nonnegative matrix $X$ and an integer $k$, the PSD factorization problem consists in finding, if possible, symmetric $k$-by-$k$ positive semidefinite matrices $\{A^1,...,A^m\}$ and $\{B^1,...,B^n\}$ such that $X_{i,j}=\text{trace}(A^iB^j)$ for $i=1,...,m$, and $j=1,...n$. PSD factorization is NP-hard. In this work, we introduce several local optimization schemes to tackle this problem: a fast projected gradient method and two algorithms based on the coordinate descent framework. The main application of PSD factorization is the computation of semidefinite extensions, that is, the representations of polyhedrons as projections of spectrahedra, for which the matrix to be factorized is the slack matrix of the polyhedron. We compare the performance of our algorithms on this class of problems. In particular, we compute the PSD extensions of size $k=1+ \lceil \log_2(n) \rceil$ for the regular $n$-gons when $n=5$, $8$ and $10$. We also show how to generalize our algorithms to compute the square root rank (which is the size of the factors in a PSD factorization where all factor matrices $A^i$ and $B^j$ have rank one) and completely PSD factorizations (which is the special case where the input matrix is symmetric and equality $A^i=B^i$ is required for all $i$).Comment: 21 pages, 3 figures, 3 table