Journal
FRONTIERS OF MATHEMATICS IN CHINA
Volume 7, Issue 2, Pages 365-384Publisher
HIGHER EDUCATION PRESS
DOI: 10.1007/s11464-012-0194-5
Keywords
nonnegative matrix factorization; matrix completion; alternating direction method; hyperspectral unmixing
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Funding
- NSF [DMS-07-48839, ECCS-1028790, DMS-0811188]
- ONR [N00014-08-1-1101]
- ARL
- ARO [W911NF-09-1-0383]
- NSF through UCLA IPAM [DMS-0439872]
- National Natural Science Foundation of China [11101274]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [0748839, 1115950] Funding Source: National Science Foundation
- Div Of Electrical, Commun & Cyber Sys
- Directorate For Engineering [1028790] Funding Source: National Science Foundation
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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.
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