Convergence analysis of the eAPG algorithm for nonnegative matrix factorization (Q2890553)

The paper deals with the nonnegative matrix factorization (NMF), i.e., for a given \((m\times n)\)-matrix \(V\), NMF finds an approximate factorization: NEWLINE\[NEWLINE V \approx WH, NEWLINE\]NEWLINE where \(W\) is an \((m\times r)\)-matrix and \(H\) is an \((r\times n)\)-matrix. The positive integer \(r\) is chosen to be smaller than \(m\) and \(n\). This problem is reformulated as minimizing the following cost function: NEWLINE\[NEWLINE f(W,H) := \|V-WH\|^2_F, NEWLINE\]NEWLINE where the Frobenius norm is used. The computation is based on the multiplicative update algorithm. Its variant called elementwisely alternating projected gradient (eAPG) algorithm is studied in the paper. The analysis starts by the proof of the existence of the equilibrium point, i.e., the point satisfying optimality conditions, and by the construction of the invariant sets. Then, it is proved that the eAPG algorithm is locally convergent. In addition, the conditions, which satisfy that the non-zero equilibrium point exists and is stable, can cause that the algorithm converges to different values. The numerical experiments illustrate the convergence of the eAPG algorithm and analyze the importance of two convergence conditions. The paper is well-written and contains the valuable contribution to the theory of the NMF.