An alternating direction algorithm for matrix completion with nonnegative factors
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 classical 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.
Annihilating Filter-Based Low-Rank Hankel Matrix Approach for Image Inpainting
In this paper, we propose a patch-based image inpainting method using a low-rank Hankel structured matrix completion approach. The proposed method exploits the annihilation property between a shift-invariant filter and image data observed in many existing inpainting algorithms. In particular, by exploiting the commutative property of the convolution, the annihilation property results in a low-rank block Hankel structure data matrix, and the image inpainting problem becomes a low-rank structured matrix completion problem. The block Hankel structured matrices are obtained patch-by-patch to adapt to the local changes in the image statistics. To solve the structured low-rank matrix completion problem, we employ an alternating direction method of multipliers with factorization matrix initialization using the low-rank matrix fitting algorithm. As a side product of the matrix factorization, locally adaptive dictionaries can be also easily constructed. Despite the simplicity of the algorithm, the experimental results using irregularly subsampled images as well as various images with globally missing patterns showed that the proposed method outperforms existing state-of-the-art image inpainting methods.
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