Minimax estimation in sparse canonical correlation analysis

TL;DR

This paper establishes the optimal rates for estimating sparse canonical correlation directions in high-dimensional data, providing theoretical guarantees and revealing that nuisance parameters and residual directions do not affect these rates.

Contribution

It offers the first theoretical minimax estimation rates for sparse canonical correlation analysis in high dimensions, including analysis of nuisance parameters and residual directions.

Findings

Minimax rates are unaffected by nuisance covariance matrices.

Residual canonical directions do not influence minimax rates under certain conditions.

Theoretical tools include a generalized sin-theta theorem and Gaussian quadratic form bounds.

Abstract

Canonical correlation analysis is a widely used multivariate statistical technique for exploring the relation between two sets of variables. This paper considers the problem of estimating the leading canonical correlation directions in high-dimensional settings. Recently, under the assumption that the leading canonical correlation directions are sparse, various procedures have been proposed for many high-dimensional applications involving massive data sets. However, there has been few theoretical justification available in the literature. In this paper, we establish rate-optimal nonasymptotic minimax estimation with respect to an appropriate loss function for a wide range of model spaces. Two interesting phenomena are observed. First, the minimax rates are not affected by the presence of nuisance parameters, namely the covariance matrices of the two sets of random variables, though they need to be estimated in the canonical correlation analysis problem. Second, we allow the presence of the residual canonical correlation directions. However, they do not influence the minimax rates under a mild condition on eigengap. A generalized sin-theta theorem and an empirical process bound for Gaussian quadratic forms under rank constraint are used to establish the minimax upper bounds, which may be of independent interest.