On Truly Block Eigensolvers via Riemannian Optimization
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Proceedings of the TwentyFirst International Conference on Artificial Intelligence and Statistics, PMLR 84:168177, 2018.
Abstract
We study theoretical properties of block solvers for the eigenvalue problem. Despite a recent surge of interest in such eigensolver analysis, truly block solvers have received relatively less attention, in contrast to the majority of studies concentrating on vector versions and nontruly block versions that rely on the deflation strategy. In fact, truly block solvers are more widely deployed in practice by virtue of its simplicity without compromise on accuracy. However, the corresponding theoretical analysis remains inadequate for firstorder solvers, as only local and kth gapdependent rates of convergence have been established thus far. This paper is devoted to revealing significantly better or asyetunknown theoretical properties of such solvers. We present a novel convergence analysis in a unified framework for three types of firstorder Riemannian solvers, i.e., deterministic, vanilla stochastic, and stochastic with variance reduction, that are to find topk eigenvectors of a real symmetric matrix, in full generality. In particular, the issue of zero gaps between eigenvalues, to the best of our knowledge for the first time, is explicitly considered for these solvers, which brings new understandings, e.g., the dependence of convergence on gaps other than the kth one. We thus propose the concept of generalized kth gap. Three types of solvers are proved to converge to a globally optimal solution at a global, generalized kth gapdependent, and linear or sublinear rate.
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