Sparse Canonical Correlation Analysis via Concave Minimization

TL;DR

This paper introduces a novel sparse CCA method with a convex formulation and a two-step algorithm, improving interpretability and computational efficiency for high-dimensional multi-view data analysis.

Contribution

The paper proposes a convex reformulation of sparse CCA, a two-step algorithm for sparsity inference, and extensions for directed and multi-view sparse CCA, enhancing interpretability and efficiency.

Findings

Superior convergence and computational efficiency demonstrated in simulations.

Accurate recovery of canonical directions and correlations.

Application to multi-omic data reveals biological associations.

Abstract

A new approach to the sparse Canonical Correlation Analysis (sCCA)is proposed with the aim of discovering interpretable associations in very high-dimensional multi-view, i.e.observations of multiple sets of variables on the same subjects, problems. Inspired by the sparse PCA approach of Journee et al. (2010), we also show that the sparse CCA formulation, while non-convex, is equivalent to a maximization program of a convex objective over a compact set for which we propose a first-order gradient method. This result helps us reduce the search space drastically to the boundaries of the set. Consequently, we propose a two-step algorithm, where we first infer the sparsity pattern of the canonical directions using our fast algorithm, then we shrink each view, i.e. observations of a set of covariates, to contain observations on the sets of covariates selected in the previous step, and compute their canonical directions via any CCA algorithm. We also introduceDirected Sparse CCA, which is able to find associations which are aligned with a specified experiment design, andMulti-View sCCA which is used to discover associations between multiple sets of covariates. Our simulations establish the superior convergence properties and computational efficiency of our algorithm as well as accuracy in terms of the canonical correlation and its ability to recover the supports of the canonical directions. We study the associations between metabolomics, trasncriptomics and microbiomics in a multi-omic study usingMuLe, which is an R-package that implements our approach, in order to form hypotheses on mechanisms of adaptations of Drosophila Melanogaster to high doses of environmental toxicants, specifically Atrazine, which is a commonly used chemical fertilizer.