On Truly Block Eigensolvers via Riemannian Optimization

Zhiqiang Xu, Xin Gao
Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, PMLR 84:168-177, 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 non-truly 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 first-order solvers, as only local and k-th gap-dependent rates of convergence have been established thus far. This paper is devoted to revealing significantly better or as-yet-unknown theoretical properties of such solvers. We present a novel convergence analysis in a unified framework for three types of first-order Riemannian solvers, i.e., deterministic, vanilla stochastic, and stochastic with variance reduction, that are to find top-k 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 k-th one. We thus propose the concept of generalized k-th gap. Three types of solvers are proved to converge to a globally optimal solution at a global, generalized k-th gap-dependent, and linear or sub-linear rate.

Cite this Paper

BibTeX
@InProceedings{pmlr-v84-xu18b, title = {On Truly Block Eigensolvers via Riemannian Optimization}, author = {Xu, Zhiqiang and Gao, Xin}, booktitle = {Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics}, pages = {168--177}, year = {2018}, editor = {Storkey, Amos and Perez-Cruz, Fernando}, volume = {84}, series = {Proceedings of Machine Learning Research}, month = {09--11 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v84/xu18b/xu18b.pdf}, url = {https://proceedings.mlr.press/v84/xu18b.html}, 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 non-truly 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 first-order solvers, as only local and k-th gap-dependent rates of convergence have been established thus far. This paper is devoted to revealing significantly better or as-yet-unknown theoretical properties of such solvers. We present a novel convergence analysis in a unified framework for three types of first-order Riemannian solvers, i.e., deterministic, vanilla stochastic, and stochastic with variance reduction, that are to find top-k 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 k-th one. We thus propose the concept of generalized k-th gap. Three types of solvers are proved to converge to a globally optimal solution at a global, generalized k-th gap-dependent, and linear or sub-linear rate.} }
Endnote
%0 Conference Paper %T On Truly Block Eigensolvers via Riemannian Optimization %A Zhiqiang Xu %A Xin Gao %B Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2018 %E Amos Storkey %E Fernando Perez-Cruz %F pmlr-v84-xu18b %I PMLR %P 168--177 %U https://proceedings.mlr.press/v84/xu18b.html %V 84 %X 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 non-truly 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 first-order solvers, as only local and k-th gap-dependent rates of convergence have been established thus far. This paper is devoted to revealing significantly better or as-yet-unknown theoretical properties of such solvers. We present a novel convergence analysis in a unified framework for three types of first-order Riemannian solvers, i.e., deterministic, vanilla stochastic, and stochastic with variance reduction, that are to find top-k 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 k-th one. We thus propose the concept of generalized k-th gap. Three types of solvers are proved to converge to a globally optimal solution at a global, generalized k-th gap-dependent, and linear or sub-linear rate.
APA
Xu, Z. & Gao, X.. (2018). On Truly Block Eigensolvers via Riemannian Optimization. Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 84:168-177 Available from https://proceedings.mlr.press/v84/xu18b.html.

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