Minimax Quasi-Bayesian estimation in sparse canonical correlation analysis via a Rayleigh quotient function

TL;DR

This paper introduces a computationally efficient quasi-Bayesian method for sparse canonical correlation analysis that achieves minimax rates and outperforms existing techniques, with applications to Covid-19 data analysis.

Contribution

It develops a novel Bayesian framework using a Rayleigh quotient function and spike-and-slab prior for sparse CCA, improving both accuracy and computational efficiency.

Findings

Outperforms state-of-the-art sparse CCA methods in simulations

Achieves minimax estimation rates theoretically

Effectively correlates clinical and proteomic data in Covid-19 study

Abstract

Canonical correlation analysis (CCA) is a popular statistical technique for exploring relationships between datasets. In recent years, the estimation of sparse canonical vectors has emerged as an important but challenging variant of the CCA problem, with widespread applications. Unfortunately, existing rate-optimal estimators for sparse canonical vectors have high computational cost. We propose a quasi-Bayesian estimation procedure that not only achieves the minimax estimation rate, but also is easy to compute by Markov Chain Monte Carlo (MCMC). The method builds on Tan et al. (2018) and uses a re-scaled Rayleigh quotient function as the quasi-log-likelihood. However, unlike Tan et al. (2018), we adopt a Bayesian framework that combines this quasi-log-likelihood with a spike-and-slab prior to regularize the inference and promote sparsity. We investigate the empirical behavior of the proposed method on both continuous and truncated data, and we demonstrate that it outperforms several state-of-the-art methods. As an application, we use the proposed methodology to maximally correlate clinical variables and proteomic data for better understanding the Covid-19 disease.