Efficient and Non-Convex Coordinate Descent for Symmetric Nonnegative Matrix Factorization

Given a symmetric nonnegative matrix A, symmetric nonnegative matrix factorization (symNMF) is the problem of finding a nonnegative matrix H, usually with much fewer columns than A, such that A ≈ HH
^T
. SymNMF can be used for data analysis and in particular for various clustering tasks. Unlike standard NMF, which is traditionally solved by a series of quadratic (convex) subproblems, we propose to solve symNMF by directly solving the nonconvex problem, namely, minimize ∥A - HH
^T
∥
²
, which is a fourth-order nonconvex problem. In this paper, we propose simple and very efficient coordinate descent schemes, which solve a series of fourth-order univariate subproblems exactly. We also derive convergence guarantees for our methods and show that they perform favorably compared to recent state-of-the-art methods on synthetic and real-world datasets, especially on large and sparse input matrices.