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The bounds of restricted isometry constants for low rank matrices recovery

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Abstract

This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 < p ⩽ 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant (RIC). Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to δ A4r < 0.558 and δ A3r < 0.4721, respectively. Recently, the upper bounds of RIC can be improved to δ A2r < 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to δ A2r < 0.4931 and δ A2r < 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with δ A2r > 1/√2 or δ A r > 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δ A2r and δ A r . Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p (0 < p < 1) quasi norm minimization problem.

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Authors and Affiliations

  1. Department of Mathematics, Shaoxing University, Shaoxing, 312000, China

    HuiMin Wang

  2. Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

    HuiMin Wang & Song Li

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  1. HuiMin Wang
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  2. Song Li
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Correspondence to Song Li.

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Wang, H., Li, S. The bounds of restricted isometry constants for low rank matrices recovery. Sci. China Math. 56, 1117–1127 (2013). https://doi.org/10.1007/s11425-013-4624-y

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  • DOI: https://doi.org/10.1007/s11425-013-4624-y

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