On reduced rank nonnegative matrix factorization for symmetric nonnegative matrices Original Research Article

Let image be a nonnegative matrix. The nonnegative matrix factorization (NNMF) problem consists of finding nonnegative matrix factors image and Hset membership, variantRr,n such that V≈WH. Lee and Seung proposed two algorithms, one of which finds nonnegative W and H such that short parallelV−WHshort parallelF 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 1less-than-or-equals, slantrless-than-or-equals, slantrank(V), a nonnegative factorization WH which coincides with the best approximation in the Frobenius norm to V in image of rank not exceeding r.