Efficient Algorithms for Large-scale Generalized Eigenvector Computation and Canonical Correlation Analysis

Abstract: This paper considers the problem of canonical-correlation analysis (CCA) (Hotelling, 1936) and, more broadly, the generalized eigenvector problem for a pair of symmetric matrices. These are two fundamental problems in data analysis and scientific computing with numerous applications in machine learning and statistics (Shi and Malik, 2000; Hardoon et al., 2004; Witten et al., 2009). We provide simple iterative algorithms, with improved runtimes, for solving these problems that are globally linearly convergent with moderate dependencies on the condition numbers and eigenvalue gaps of the matrices involved. We obtain our results by reducing CCA to the top-$k$ generalized eigenvector problem. We solve this problem through a general framework that simply requires black box access to an approximate linear system solver. Instantiating this framework with accelerated gradient descent we obtain a running time of $O(\frac{z k \sqrt{\kappa}}{\rho} \log(1/\epsilon) \log \left(k\kappa/\rho\right))$ where $z$ is the total number of nonzero entries, $\kappa$ is the condition number and $\rho$ is the relative eigenvalue gap of the appropriate matrices. Our algorithm is linear in the input size and the number of components $k$ up to a $\log(k)$ factor. This is essential for handling large-scale matrices that appear in practice. To the best of our knowledge this is the first such algorithm with global linear convergence. We hope that our results prompt further research and ultimately improve the practical running time for performing these important data analysis procedures on large data sets.