$\ell_0$-based Sparse Canonical Correlation Analysis

TL;DR

This paper introduces $ ext{ extonehalf}$-based sparse CCA methods that select relevant variables for better correlated representations, especially in high-dimensional and multi-modal data, using stochastic gates and deep nets.

Contribution

The paper proposes $ ext{ extonehalf}$-CCA and $ ext{ extonehalf}$-Deep CCA, novel sparse CCA models that incorporate stochastic gating and deep learning to improve variable selection and correlation analysis.

Findings

Outperforms existing linear and non-linear CCA models in synthetic and real data.

Effectively gates nuisance variables, enhancing the quality of extracted representations.

Demonstrates improved correlation detection in high-dimensional, multi-modal datasets.

Abstract

Canonical Correlation Analysis (CCA) models are powerful for studying the associations between two sets of variables. The canonically correlated representations, termed \textit{canonical variates} are widely used in unsupervised learning to analyze unlabeled multi-modal registered datasets. Despite their success, CCA models may break (or overfit) if the number of variables in either of the modalities exceeds the number of samples. Moreover, often a significant fraction of the variables measures modality-specific information, and thus removing them is beneficial for identifying the \textit{canonically correlated variates}. Here, we propose $\ell_0$-CCA, a method for learning correlated representations based on sparse subsets of variables from two observed modalities. Sparsity is obtained by multiplying the input variables by stochastic gates, whose parameters are learned together with the CCA weights via an $\ell_0$-regularized correlation loss. We further propose $\ell_0$-Deep CCA for solving the problem of non-linear sparse CCA by modeling the correlated representations using deep nets. We demonstrate the efficacy of the method using several synthetic and real examples. Most notably, by gating nuisance input variables, our approach improves the extracted representations compared to other linear, non-linear and sparse CCA-based models.