A simple and provable algorithm for sparse diagonal CCA
This addresses the challenge of efficiently extracting sparse, interpretable correlations in high-dimensional data for fields like neuroimaging, though it is incremental as it builds on existing sparse CCA methods with added theoretical guarantees.
The authors tackled the NP-hard problem of sparse Canonical Correlation Analysis (CCA) by proposing a combinatorial algorithm for sparse diagonal CCA, which provides theoretical approximation guarantees and scales linearly with input variables, achieving precise sparsity control as demonstrated in a neuroimaging application.
Given two sets of variables, derived from a common set of samples, sparse Canonical Correlation Analysis (CCA) seeks linear combinations of a small number of variables in each set, such that the induced canonical variables are maximally correlated. Sparse CCA is NP-hard. We propose a novel combinatorial algorithm for sparse diagonal CCA, i.e., sparse CCA under the additional assumption that variables within each set are standardized and uncorrelated. Our algorithm operates on a low rank approximation of the input data and its computational complexity scales linearly with the number of input variables. It is simple to implement, and parallelizable. In contrast to most existing approaches, our algorithm administers precise control on the sparsity of the extracted canonical vectors, and comes with theoretical data-dependent global approximation guarantees, that hinge on the spectrum of the input data. Finally, it can be straightforwardly adapted to other constrained variants of CCA enforcing structure beyond sparsity. We empirically evaluate the proposed scheme and apply it on a real neuroimaging dataset to investigate associations between brain activity and behavior measurements.