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A perturbation inequality for concave functions of singular values and its applications in low-rank matrix recovery. (English) Zbl 1335.15031
Summary: In this paper, we establish the following perturbation result concerning the singular values of a matrix: Let $$A, B \in \mathbb{R}^{m \times n}$$ be given matrices, and let $$f : \mathbb{R}_+ \to \mathbb{R}_+$$ be a concave function satisfying $$f(0) = 0$$. Then, we have $\sum_{i = 1}^{\min \{m, n \}} | f(\sigma_i(A)) - f(\sigma_i(B)) | \leq \sum_{i = 1}^{\min \{m, n \}} f(\sigma_i(A - B)),$ where $$\sigma_i(\cdot)$$ denotes the $$i$$-th largest singular value of a matrix. This answers an open question that is of interest to both the compressive sensing and linear algebra communities. In particular, by taking $$f(\cdot) = (\cdot)^p$$ for any $$p \in(0, 1]$$, we obtain a perturbation inequality for the so-called Schatten $$p$$-quasi-norm, which allows us to confirm the validity of a number of previously conjectured conditions for the recovery of low-rank matrices via the popular Schatten $$p$$-quasi-norm heuristic. We believe that our result will find further applications, especially in the study of low-rank matrix recovery.

##### MSC:
 15A45 Miscellaneous inequalities involving matrices 94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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