Quadratic nonnegative matrix factorization

In Nonnegative Matrix Factorization (NMF), a nonnegative matrix is approximated by a product of lower-rank factorizing matrices. Most NMF methods assume that each factorizing matrix appears only once in the approximation, thus the approximation is linear in the factorizing matrices. We present a new class of approximative NMF methods, called Quadratic Nonnegative Matrix Factorization (QNMF), where some factorizing matrices occur twice in the approximation. We demonstrate QNMF solutions to four potential pattern recognition problems in graph partitioning, two-way clustering, estimating hidden Markov chains, and graph matching. We derive multiplicative algorithms that monotonically decrease the approximation error under a variety of measures. We also present extensions in which one of the factorizing matrices is constrained to be orthogonal or stochastic. Empirical studies show that for certain application scenarios, QNMF is more advantageous than other existing nonnegative matrix factorization methods.

► We present a new class of approximative nonnegative matrix factorization methods.
► The approximation in this class is quadratic to some factorizing matrices.
► We show how to derive convergent multiplicative algorithms for various divergences.
► Our method can accommodate the stochasticity or orthogonality constraint.
► The proposed method is more advantageous in four pattern recognition problems.