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Reconstruction of a low-rank matrix in the presence of Gaussian noise

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  • Shabalin, Andrey A.
  • Nobel, Andrew B.

Abstract

This paper addresses the problem of reconstructing a low-rank signal matrix observed with additive Gaussian noise. We first establish that, under mild assumptions, one can restrict attention to orthogonally equivariant reconstruction methods, which act only on the singular values of the observed matrix and do not affect its singular vectors. Using recent results in random matrix theory, we then propose a new reconstruction method that aims to reverse the effect of the noise on the singular value decomposition of the signal matrix. In conjunction with the proposed reconstruction method we also introduce a Kolmogorov–Smirnov based estimator of the noise variance.

Suggested Citation

  • Shabalin, Andrey A. & Nobel, Andrew B., 2013. "Reconstruction of a low-rank matrix in the presence of Gaussian noise," Journal of Multivariate Analysis, Elsevier, vol. 118(C), pages 67-76.
    • Handle: RePEc:eee:jmvana:v:118:y:2013:i:c:p:67-76
    DOI: 10.1016/j.jmva.2013.03.005
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    References listed on IDEAS

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    1. L. Györfi & I. Vajda & E. Meulen, 1996. "Minimum kolmogorov distance estimates of parameters and parametrized distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 43(1), pages 237-255, December.
    2. Baik, Jinho & Silverstein, Jack W., 2006. "Eigenvalues of large sample covariance matrices of spiked population models," Journal of Multivariate Analysis, Elsevier, vol. 97(6), pages 1382-1408, July.
    3. Yata, Kazuyoshi & Aoshima, Makoto, 2012. "Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 193-215.
    4. Dozier, R. Brent & Silverstein, Jack W., 2007. "On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices," Journal of Multivariate Analysis, Elsevier, vol. 98(4), pages 678-694, April.
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    Cited by:

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    2. Junhui Cai & Dan Yang & Wu Zhu & Haipeng Shen & Linda Zhao, 2021. "Network regression and supervised centrality estimation," Papers 2111.12921, arXiv.org.
    3. Li, Gen & Yang, Dan & Nobel, Andrew B. & Shen, Haipeng, 2016. "Supervised singular value decomposition and its asymptotic properties," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 7-17.
    4. Leeb, William, 2022. "Optimal singular value shrinkage for operator norm loss: Extending to non-square matrices," Statistics & Probability Letters, Elsevier, vol. 186(C).
    5. Gen Li & Sungkyu Jung, 2017. "Incorporating covariates into integrated factor analysis of multi‐view data," Biometrics, The International Biometric Society, vol. 73(4), pages 1433-1442, December.
    6. Bigot, Jérémie & Deledalle, Charles, 2022. "Low-rank matrix denoising for count data using unbiased Kullback-Leibler risk estimation," Computational Statistics & Data Analysis, Elsevier, vol. 169(C).
    7. Gong, Tingnan & Zhang, Weiping & Chen, Yu, 2023. "Uncovering block structures in large rectangular matrices," Journal of Multivariate Analysis, Elsevier, vol. 198(C).

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