Doubly Accelerated Methods for Faster CCA and Generalized Eigendecomposition

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Zeyuan Allen-Zhu, Yuanzhi Li ;
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:98-106, 2017.

Abstract

We study k-GenEV, the problem of finding the top k generalized eigenvectors, and k-CCA, the problem of finding the top k vectors in canonical-correlation analysis. We propose algorithms LazyEV and LazyCCA to solve the two problems with running times linearly dependent on the input size and on k. Furthermore, our algorithms are doubly-accelerated: our running times depend only on the square root of the matrix condition number, and on the square root of the eigengap. This is the first such result for both k-GenEV or k-CCA. We also provide the first gap-free results, which provide running times that depend on $1/\sqrt{\varepsilon}$ rather than the eigengap.

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