  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.
