Low-Dimensional Polytope Approximation and Its Applications to Nonnegative Matrix Factorization

Abstract

In this study, nonnegative matrix factorization is recast as the problem of approximating a polytope on the probability simplex by another polytope with fewer facets. Working on the probability simplex has the advantage that data are limited to a compact set with a known boundary, making it easier to trace the approximation procedure. In particular, the supporting hyperplane that separates a point from a disjoint polytope, a fact asserted by the Hahn-Banach theorem, can be calculated in finitely many steps. This approach leads to a convenient way of computing the proximity map which, in contrast to most existing algorithms where only an approximate map is used, finds the unique and global minimum per iteration. This paper sets up a theoretical framework, outlines a numerical algorithm, and suggests an effective implementation. Testing results strongly evidence that this approach obtains a better low rank nonnegative matrix approximation in fewer steps than conventional methods. Introduction. The problem of nonnegative matrix factorization (NMF) arisesin a large variety of disciplines in sciences and engineering. Its wide range of important applications such as text mining, cheminformatics, factor retrieval, image articulation, dimension reduction, and so on has attracted considerable research efforts. Many different kinds of NMF techniques have been proposed in the literature, notably the popular Lee and Seung iterative update algorithm [17,18]. Successful applications to various models with nonnegative data values are abounding. We can hardly be exhaustive by suggesting [6,11,12,13,16,20,19,21,22,24], and the references contained therein as a partial list of interesting work. The review article [27] contains many other more recent applications and references.The basic idea behind NMF is the typical linear model

Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework

“…Several approaches for solving the NLS subproblems proposed in NMF literature are discussed in Sect. 4 [18,42,51,53,59,71]. According to Theorem 1, the convergence property of the ANLS framework can be stated as follows.…”

“…The convergence property of the scalar block case is similar to that of the vector block case. (18) are attained at each step, every limit point of the sequence (W, H) (i) generated by the BCD method with K (M + N ) scalar blocks is a stationary point of (2).…”

“…In the ANLS method, the subproblems appear as the nonnegativity constrained least squares (NLS) problems. Much research has been devoted to design NMF algorithms based on efficient methods to solve the NLS subproblems [18,42,51,53,59,71]. A review of many successful algorithms for the NLS subproblems is provided in Sect.…”

Regularized nonnegative matrix factorization: Geometrical interpretation and application to spectral unmixing

“…The points in P S (I−1) (Y ) form the convex polytope C(Y ) (Chu and Lin, 2008). If the matrix X is sufficiently sparse (see Definition 2), the vertices of C(Y ) correspond to those column vectors of A that intersect with S (I−1) .…”

Nonnegative Matrix Factorizations Performing Object Detection and Localization

“…It has been proved that the above algorithm converges into a local minimum. Other techniques, such as alternating nonnegative least squares method or bound-constrained optimization algorithms, such as projected gradient method, have also been used when additional constraints are added to the nonnegativity of the matrices W or H [34][35][36].…”