Canonical Correlation Analysis as Reduced Rank Regression in High Dimensions

TL;DR

This paper proposes a novel approach to high-dimensional canonical correlation analysis by framing it as a reduced rank regression problem, enhancing scalability and adaptability over existing sparse methods.

Contribution

It introduces a reduced rank regression framework for CCA in high dimensions, improving computational efficiency and flexibility with structural priors.

Findings

Effective in very high-dimensional settings

Maintains accuracy and computational efficiency

Validated on simulated and real datasets

Abstract

Canonical Correlation Analysis (CCA) is a widespread technique for discovering linear relationships between two sets of variables $X \in \mathbb{R}^{n \times p}$ and $Y \in \mathbb{R}^{n \times q}$. In high dimensions however, standard estimates of the canonical directions cease to be consistent without assuming further structure. In this setting, a possible solution consists in leveraging the presumed sparsity of the solution: only a subset of the covariates span the canonical directions. While the last decade has seen a proliferation of sparse CCA methods, practical challenges regarding the scalability and adaptability of these methods still persist. To circumvent these issues, this paper suggests an alternative strategy that uses reduced rank regression to estimate the canonical directions when one of the datasets is high-dimensional while the other remains low-dimensional. By casting the problem of estimating the canonical direction as a regression problem, our estimator is able to leverage the rich statistics literature on high-dimensional regression and is easily adaptable to accommodate a wider range of structural priors. Our proposed solution maintains computational efficiency and accuracy, even in the presence of very high-dimensional data. We validate the benefits of our approach through a series of simulated experiments and further illustrate its practicality by applying it to three real-world datasets.