On Support Recovery with Sparse CCA: Information Theoretic and Computational Limits

TL;DR

This paper investigates the limits of support recovery in high-dimensional sparse Canonical Correlation Analysis, identifying regimes where recovery is possible, impossible, or computationally hard, and providing theoretical and empirical insights.

Contribution

It characterizes the regimes of sparsity where support recovery is feasible, impossible, or computationally hard, and introduces analysis of simple and thresholding methods in these regimes.

Findings

Support recovery is easy in low sparsity regimes.

Support recovery is information theoretically impossible in high sparsity regimes.

Polynomial time support recovery is feasible in moderate sparsity but becomes hard in higher moderate sparsity.

Abstract

In this paper we consider asymptotically exact support recovery in the context of high dimensional and sparse Canonical Correlation Analysis (CCA). Our main results describe four regimes of interest based on information theoretic and computational considerations. In regimes of "low" sparsity we describe a simple, general, and computationally easy method for support recovery, whereas in a regime of "high" sparsity, it turns out that support recovery is information theoretically impossible. For the sake of information theoretic lower bounds, our results also demonstrate a non-trivial requirement on the "minimal" size of the non-zero elements of the canonical vectors that is required for asymptotically consistent support recovery. Subsequently, the regime of "moderate" sparsity is further divided into two sub-regimes. In the lower of the two sparsity regimes, using a sharp analysis of a coordinate thresholding (Deshpande and Montanari, 2014) type method, we show that polynomial time support recovery is possible. In contrast, in the higher end of the moderate sparsity regime, appealing to the "Low Degree Polynomial" Conjecture (Kunisky et al., 2019), we provide evidence that polynomial time support recovery methods are inconsistent. Finally, we carry out numerical experiments to compare the efficacy of various methods discussed.