## The Annals of Statistics

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

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

#### Abstract

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

**Source**

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

**Dates**

Received: May 2016

Revised: November 2016

First available in Project Euclid: 22 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1519268424

**Digital Object Identifier**

doi:10.1214/17-AOS1541

**Mathematical Reviews number (MathSciNet)**

MR3766946

**Zentralblatt MATH identifier**

06865105

**Subjects**

Primary: 62H12: Estimation 62C20: Minimax procedures

Secondary: 62H25: Factor analysis and principal components; correspondence analysis

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1519268424

#### 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.Digital Object Identifier: doi:10.1214/17-AOS1541SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.