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This paper considers the problem of canonical-correlation analysis (CCA)(Hotelling, 1936) and, more broadly, the generalized eigenvector problem for apair of symmetric matrices. These are two fundamental problems in data analysisand scientific computing with numerous applications in machine learning andstatistics (Shi and Malik, 2000; Hardoon et al., 2004; Witten et al., 2009).  We provide simple iterative algorithms, with improved runtimes, for solvingthese problems that are globally linearly convergent with moderate dependencieson the condition numbers and eigenvalue gaps of the matrices involved.  We obtain our results by reducing CCA to the top-$k$ generalized eigenvectorproblem. We solve this problem through a general framework that simply requiresblack box access to an approximate linear system solver. Instantiating thisframework 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 ofthe appropriate matrices.  Our algorithm is linear in the input size and the number of components $k$ upto a $\log(k)$ factor. This is essential for handling large-scale matrices thatappear in practice. To the best of our knowledge this is the first suchalgorithm with global linear convergence. We hope that our results promptfurther research and ultimately improve the practical running time forperforming these important data analysis procedures on large data sets.

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