An alternating direction algorithm for matrix completion with nonnegative factors

Abstract

This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices X and Y so that the product XY approximates a nonnegative data matrix M whose elements are partially known (to a certain accuracy). This problem aggregates two existing problems: (i) nonnegative matrix factorization where all entries of M are given, and (ii) low-rank matrix completion where nonnegativity is not required. By taking the advantages of both nonnegativity and low-rankness, one can generally obtain superior results than those of just using one of the two properties. We propose to solve the non-convex constrained least-squares problem using an algorithm based on the classic alternating direction augmented Lagrangian method. Preliminary convergence properties of the algorithm and numerical simulation results are presented. Compared to a recent algorithm for nonnegative matrix factorization, the proposed algorithm produces factorizations of similar quality using only about half of the matrix entries. On tasks of recovering incomplete grayscale and hyperspectral images, the proposed algorithm yields overall better qualities than those produced by two recent matrix-completion algorithms that do not exploit nonnegativity.

Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm

“…Numerical simulations shows that the simple approach in subsection 2.1, though being very reliable, is not efficient on large yet very low-rank matrices. A possible acceleration technique may involve applying an extension of the classic augmented-Lagrangian-based alternating direction method (ADM) for convex optimization to the factorization model (see [29,35,39] for such ADM extensions). However, in this paper, we investigate a nonlinear Successive Over-Relaxation (SOR) approach that we found to be particularly effective for solving the matrix completion problem.…”

Compressive Hyperspectral Imaging and Anomaly Detection

“…We compared our algorithm in [9] to the algorithm proposed in [10], which takes complete samples of M and performs similar ADM-based iterations on random matrices with varying number of sampled entries. The recovery qualities and speeds are illustrated in Figure 23.…”

“…We compared our algorithm in [9] with LMaFit [11] and FPCA [12] on recovering three-dimensional hyperspectral images from their incomplete observations. Our test hyperspectral datacube has 163 slices, and the size of each slice is 80 × 80.…”

“…We also compare the algorithm APG-MC and the ADM-based algorithm (ADM-MC) in [9] on the hyperspectral data used in last test. It is demonstrated in [9] that ADM-MC outperforms matrix completion solvers APGL and LMaFit for nonnegative matrix completion problems, because ADM-MC utilizes the nonnegativity property of the data while the latter two do not.…”

Flexible Nonparametric Kernel Learning with Different Loss Functions

“…X and Y, and so we can only consider local convergence [3]. As in [13], we show below a necessary condition for local convergence.…”