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Robustness properties of dimensionality reduction with Gaussian random matrices

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Abstract

In this paper, motivated by the results in compressive phase retrieval, we study the robustness properties of dimensionality reduction with Gaussian random matrices having arbitrarily erased rows. We first study the robustness property against erasure for the almost norm preservation property of Gaussian random matrices by obtaining the optimal estimate of the erasure ratio for a small given norm distortion rate. As a consequence, we establish the robustness property of Johnson-Lindenstrauss lemma and the robustness property of restricted isometry property with corruption for Gaussian random matrices. Secondly, we obtain a sharp estimate for the optimal lower and upper bounds of norm distortion rates of Gaussian random matrices under a given erasure ratio. This allows us to establish the strong restricted isometry property with the almost optimal restricted isometry property (RIP) constants, which plays a central role in the study of phaseless compressed sensing. As a byproduct of our results, we also establish the robustness property of Gaussian random finite frames under erasure.

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Acknowledgements

This work was supported by Natural Sciences and Engineering Research Council of Canada (Grant No. 05865), National Natural Science Foundation of China (Grant Nos. 11422113, 91630203, 11021101 and 11331012) and National Basic Research Program of China (973 Program) (Grant No. 2015CB856000). The authors are thankful to Yang Wang for informing them the reference [24] and its connection with their work.

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Correspondence to ZhiQiang Xu.

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Han, B., Xu, Z. Robustness properties of dimensionality reduction with Gaussian random matrices. Sci. China Math. 60, 1753–1778 (2017). https://doi.org/10.1007/s11425-016-9018-x

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  • DOI: https://doi.org/10.1007/s11425-016-9018-x

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