Singular value problems under nonnegativity constraints

Abstract

Let A be a real matrix of size mxn. In classical linear algebra, a real number sigma is called a singular value of A if there exist unit vectors u is an element of Rm and v is an element of Rn such that Av=sigma u and A inverted perpendicular u=sigma v. In variational analysis, a singular value of A is viewed as a critical value of the bilinear form u,Av when u and v range on the unit spheres of Rm and Rn, respectively. If u and v are further required to be nonnegative, then the idea of criticality is expressed by means of a pair of complementarity problems, namely, 0 <= u perpendicular to (Av-sigma u)>= 0 and 0 <= v perpendicular to (A inverted perpendicular u-sigma v)>= 0. The parameter sigma is now called a Pareto singular value of A. In this work we study the concept of Pareto singular value and, by way of application, we analyze a problem of nonnegative matrix factorization. The set Xi (A) of Pareto singular values of A is nonempty and finite. We derive an explicit formula for the maximum number of Pareto singular values in a matrix of prescribed size. The elements of Xi (A) can be found by solving a collection of classical singular value problems involving the principal submatrices of A. Unfortunately, such a method is cost prohibitive if m and n are large. For matrices of large size we propose an algorithm of alternating minimization type. This work is a continuation of our paper entitled Cone-constrained singular value problems published in the Journal of Convex Analysis (30, 2023, pp. 1285-1306).