Dropping Symmetry for Fast Symmetric Nonnegative Matrix Factorization

Abstract

Symmetric nonnegative matrix factorization (NMF)-a special but important class of the general NMF-is demonstrated to be useful for data analysis and in particular for various clustering tasks. Unfortunately, designing fast algorithms for Symmetric NMF is not as easy as for the nonsymmetric counterpart, the later admitting the splitting property that allows efficient alternating-type algorithms.To overcome this issue, we transfer the symmetric NMF to a nonsymmetric one, then we can adopt the idea from the state-of-the-art algorithms for nonsymmetric NMF to design fast algorithms solving symmetric NMF. We rigorously establish that solving nonsymmetric reformulation returns a solution for symmetric NMF and then apply fast alternating based algorithms for the corresponding reformulated problem. Furthermore, we show these fast algorithms admit strong convergence guarantee in the sense that the generated sequence is convergent at least at a sublinear rate and it converges globally to a critical point of the symmetric NMF. We conduct experiments on both synthetic data and image clustering to support our result.

Robust Principal Component Analysis: A Construction Error Minimization Perspective

“…The above equation tells that if we set t ≤ 1 L , then the objective in Eq. ( 18) will monotonically decrease [33,34]. We now turn to find the optimal solution for W + .…”