Structured Low-Rank Approximation: Optimization on Matrix Manifold Approach

We deal with the problem to compute the nearest Structured Low-Rank Approximation (SLRA) to a given matrix in this paper. This problem arises in many practical applications, such as computation of approximate GCD of polynomials, matrix completion problems, image processing and control theory etc. We...

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Bibliographic Details
Published in:International journal of applied and computational mathematics 2021-12, Vol.7 (6), Article 242
Main Authors: Saha, Tanay, Khare, Swanand
Format: Article
Language:English
Subjects:
Applications of Mathematics
Applied mathematics
Approximation
Computational mathematics
Computational Science and Engineering
Control theory
Image processing
Manifolds (mathematics)
Mathematical analysis
Mathematical and Computational Physics
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Nuclear Energy
Operations Research/Decision Theory
Optimization
Original Paper
Polynomials
Theoretical
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Summary:We deal with the problem to compute the nearest Structured Low-Rank Approximation (SLRA) to a given matrix in this paper. This problem arises in many practical applications, such as computation of approximate GCD of polynomials, matrix completion problems, image processing and control theory etc. We reformulate this problem as an unconstrained optimization problem on an appropriately chosen Stiefel manifold. This proposed formulation is based on the condition that the computed nearest SLRA is required to have a suitable number of linearly independent vectors in its nullspace. Thus, this formulation helps in computing the nearest SLRA with exact required rank as opposed to many numerical techniques available in literature. Two different techniques to solve the optimization problem on the Stiefel manifold are discussed in this paper with several numerical case studies. The effectiveness of the proposed approach is demonstrated through numerical examples and by comparison with the benchmark methods available in the literature.
ISSN:2349-5103
2199-5796
DOI:10.1007/s40819-021-01162-8