On reduced rank nonnegative matrix factorization for symmetric nonnegative matrices

Let V is an element of R(m,n) be a nonnegative matrix. The nonnegative matrix factorization (NNMF) problem consists of finding nonnegative matrix factors W E R and H E R(r,n), such that V approximate to WH. Lee and Seung proposed two algorithms, one of which finds nonnegative W and H such that parallel toV - WHparallel to(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 in = 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 I less than or equal to r less than or equal to 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. (C) 2004 Published by Elsevier Inc.