Perturbation expansions and error bounds for the truncated singular value decomposition
| Autor: | Evgenia Chunikhina, Raviv Raich, Trung Kien Vu |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Numerical Analysis
Algebra and Number Theory Rank (linear algebra) Operator (physics) 010102 general mathematics Mathematics - Statistics Theory Numerical Analysis (math.NA) Statistics Theory (math.ST) 010103 numerical & computational mathematics 01 natural sciences Linear subspace 15A06 (Primary) 65F06 (Secondary) Matrix (mathematics) Singular value 2 × 2 real matrices Singular value decomposition FOS: Mathematics Discrete Mathematics and Combinatorics Applied mathematics Mathematics - Numerical Analysis Geometry and Topology Truncation (statistics) 0101 mathematics Mathematics |
| Zdroj: | Linear Algebra and its Applications. 627:94-139 |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/j.laa.2021.05.020 |
| Popis: | Truncated singular value decomposition is a reduced version of the singular value decomposition in which only a few largest singular values are retained. This paper presents a novel perturbation analysis for the truncated singular value decomposition for real matrices. First, we describe perturbation expansions for the singular value truncation of order $r$. We extend perturbation results for the singular subspace decomposition to derive the first-order perturbation expansion of the truncated operator about a matrix with rank greater than or equal to $r$. Observing that the first-order expansion can be greatly simplified when the matrix has exact rank $r$, we further show that the singular value truncation admits a simple second-order perturbation expansion about a rank-$r$ matrix. Second, we introduce the first-known error bound on the linear approximation of the truncated singular value decomposition of a perturbed rank-$r$ matrix. Our bound only depends on the least singular value of the unperturbed matrix and the norm of the perturbation matrix. Intriguingly, while the singular subspaces are known to be extremely sensitive to additive noises, the newly established error bound holds universally for perturbations with arbitrary magnitude. Finally, we demonstrate an application of our results to the analysis of the mean squared error associated with the TSVD-based matrix denoising solution. Comment: Accepted to Linear Algebra and Its Applications |
| Databáze: | OpenAIRE |
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