The Annals of Statistics

Rate-optimal perturbation bounds for singular subspaces with applications to high-dimensional statistics

T. Tony Cai and Anru Zhang

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Perturbation bounds for singular spaces, in particular Wedin’s $\mathop{\mathrm{sin}}\nolimits \Theta$ theorem, are a fundamental tool in many fields including high-dimensional statistics, machine learning and applied mathematics. In this paper, we establish separate perturbation bounds, measured in both spectral and Frobenius $\mathop{\mathrm{sin}}\nolimits \Theta$ distances, for the left and right singular subspaces. Lower bounds, which show that the individual perturbation bounds are rate-optimal, are also given.

The new perturbation bounds are applicable to a wide range of problems. In this paper, we consider in detail applications to low-rank matrix denoising and singular space estimation, high-dimensional clustering and canonical correlation analysis (CCA). In particular, separate matching upper and lower bounds are obtained for estimating the left and right singular spaces. To the best of our knowledge, this is the first result that gives different optimal rates for the left and right singular spaces under the same perturbation.

Article information

Ann. Statist., Volume 46, Number 1 (2018), 60-89.

Received: May 2016
Revised: November 2016
First available in Project Euclid: 22 February 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation 62C20: Minimax procedures
Secondary: 62H25: Factor analysis and principal components; correspondence analysis

Canonical correlation analysis clustering high-dimensional statistics low-rank matrix denoising perturbation bound singular value decomposition $\mathop{\mathrm{sin}}\nolimits \Theta$ distances spectral method


Cai, T. Tony; Zhang, Anru. Rate-optimal perturbation bounds for singular subspaces with applications to high-dimensional statistics. Ann. Statist. 46 (2018), no. 1, 60--89. doi:10.1214/17-AOS1541.

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Supplemental materials

  • Supplement to “Rate-optimal perturbation bounds for singular subspaces with applications to high-dimensional statistics”. The supplementary material includes the proofs for Theorem 2, Corollary 1, matrix denoising, high-dimensional clustering, canonical correlation analysis and all the technical lemmas.