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February 2020 Subspace perspective on canonical correlation analysis: Dimension reduction and minimax rates
Zhuang Ma, Xiaodong Li
Bernoulli 26(1): 432-470 (February 2020). DOI: 10.3150/19-BEJ1131

Abstract

Canonical correlation analysis (CCA) is a fundamental statistical tool for exploring the correlation structure between two sets of random variables. In this paper, motivated by the recent success of applying CCA to learn low dimensional representations of high dimensional objects, we propose two losses based on the principal angles between the model spaces spanned by the sample canonical variates and their population correspondents, respectively. We further characterize the non-asymptotic error bounds for the estimation risks under the proposed error metrics, which reveal how the performance of sample CCA depends adaptively on key quantities including the dimensions, the sample size, the condition number of the covariance matrices and particularly the population canonical correlation coefficients. The optimality of our uniform upper bounds is also justified by lower-bound analysis based on stringent and localized parameter spaces. To the best of our knowledge, for the first time our paper separates $p_{1}$ and $p_{2}$ for the first order term in the upper bounds without assuming the residual correlations are zeros. More significantly, our paper derives $(1-\lambda_{k}^{2})(1-\lambda_{k+1}^{2})/(\lambda_{k}-\lambda_{k+1})^{2}$ for the first time in the non-asymptotic CCA estimation convergence rates, which is essential to understand the behavior of CCA when the leading canonical correlation coefficients are close to $1$.

Citation

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Zhuang Ma. Xiaodong Li. "Subspace perspective on canonical correlation analysis: Dimension reduction and minimax rates." Bernoulli 26 (1) 432 - 470, February 2020. https://doi.org/10.3150/19-BEJ1131

Information

Received: 1 February 2018; Revised: 1 December 2018; Published: February 2020
First available in Project Euclid: 26 November 2019

zbMATH: 07140505
MathSciNet: MR4036040
Digital Object Identifier: 10.3150/19-BEJ1131

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 1 • February 2020
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