On Reduced Rank Nonnegative Matrix Factorization for Symmetric Nonnegative Matrices

Let V ∈ R be a nonnegative matrix. The nonnegative matrix factorization (NNMF) problem consists of finding nonnegative matrix factors W ∈ R and H ∈ R such that V ≈ WH. Lee and Seung proposed two algorithms which find 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 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. We further mention that a vehicle for our results here is the Khun–Tucker conditions. ∗Research was supported by a Rackham Fellowship of the University of Michigan. †This author’s research was supported in part by NSF grant No. DMS0201333. ‡His research was supported in part by the Air Force Office of Scientific Research under grant F49620– 02–1–0107, and by the Army Research Office under grant DAAD19–00–1–0540.