On reduced rank nonnegative matrix factorization for symmetric nonnegative matrices

Abstract

Let V∈ R m,n be a nonnegative matrix. The nonnegative matrix factorization (NNMF) problem consists of finding nonnegative matrix factors W∈ R m,r and H∈ R r, n such that V≈ WH. Lee and Seung proposed two algorithms, one of which finds nonnegative W and H such that ∥ V− WH∥ F is minimized. After examining the case in which r=1 about which a complete characterization of the solution is possible, we consider the case in which m= n and V is symmetric. We focus on questions concerning when the best approximate factorization results in the product WH being symmetric and on cases in which the best approximation cannot be a symmetric matrix. Finally, we show that the class of positive semidefinite symmetric nonnegative matrices V generated via a Soules basis admit for every 1⩽ r⩽rank( V), a nonnegative factorization WH which coincides with the best approximation in the Frobenius norm to V in R n,n of rank not exceeding r. An example of applications in which NNMF factorizations for nonnegative symmetric matrices V arise is video and other media summarization technology where V is obtained from a distance matrix.