Distributed Estimation of Generalized Matrix Rank: Efficient Algorithms and Lower Bounds

Yuchen Zhang, Martin Wainwright, Michael Jordan
Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:457-465, 2015.

Abstract

We study the following generalized matrix rank estimation problem: given an n-by-n matrix and a constant c > 0, estimate the number of eigenvalues that are greater than c. In the distributed setting, the matrix of interest is the sum of m matrices held by separate machines. We show that any deterministic algorithm solving this problem must communicate Ω(n^2) bits, which is order-equivalent to transmitting the whole matrix. In contrast, we propose a randomized algorithm that communicates only O(n) bits. The upper bound is matched by an Ω(n) lower bound on the randomized communication complexity. We demonstrate the practical effectiveness of the proposed algorithm with some numerical experiments.

Cite this Paper

BibTeX
@InProceedings{pmlr-v37-zhangc15, title = {Distributed Estimation of Generalized Matrix Rank: Efficient Algorithms and Lower Bounds}, author = {Zhang, Yuchen and Wainwright, Martin and Jordan, Michael}, booktitle = {Proceedings of the 32nd International Conference on Machine Learning}, pages = {457--465}, year = {2015}, editor = {Bach, Francis and Blei, David}, volume = {37}, series = {Proceedings of Machine Learning Research}, address = {Lille, France}, month = {07--09 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v37/zhangc15.pdf}, url = {https://proceedings.mlr.press/v37/zhangc15.html}, abstract = {We study the following generalized matrix rank estimation problem: given an n-by-n matrix and a constant c > 0, estimate the number of eigenvalues that are greater than c. In the distributed setting, the matrix of interest is the sum of m matrices held by separate machines. We show that any deterministic algorithm solving this problem must communicate Ω(n^2) bits, which is order-equivalent to transmitting the whole matrix. In contrast, we propose a randomized algorithm that communicates only O(n) bits. The upper bound is matched by an Ω(n) lower bound on the randomized communication complexity. We demonstrate the practical effectiveness of the proposed algorithm with some numerical experiments.} }
Endnote
%0 Conference Paper %T Distributed Estimation of Generalized Matrix Rank: Efficient Algorithms and Lower Bounds %A Yuchen Zhang %A Martin Wainwright %A Michael Jordan %B Proceedings of the 32nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2015 %E Francis Bach %E David Blei %F pmlr-v37-zhangc15 %I PMLR %P 457--465 %U https://proceedings.mlr.press/v37/zhangc15.html %V 37 %X We study the following generalized matrix rank estimation problem: given an n-by-n matrix and a constant c > 0, estimate the number of eigenvalues that are greater than c. In the distributed setting, the matrix of interest is the sum of m matrices held by separate machines. We show that any deterministic algorithm solving this problem must communicate Ω(n^2) bits, which is order-equivalent to transmitting the whole matrix. In contrast, we propose a randomized algorithm that communicates only O(n) bits. The upper bound is matched by an Ω(n) lower bound on the randomized communication complexity. We demonstrate the practical effectiveness of the proposed algorithm with some numerical experiments.
RIS
TY - CPAPER TI - Distributed Estimation of Generalized Matrix Rank: Efficient Algorithms and Lower Bounds AU - Yuchen Zhang AU - Martin Wainwright AU - Michael Jordan BT - Proceedings of the 32nd International Conference on Machine Learning DA - 2015/06/01 ED - Francis Bach ED - David Blei ID - pmlr-v37-zhangc15 PB - PMLR DP - Proceedings of Machine Learning Research VL - 37 SP - 457 EP - 465 L1 - http://proceedings.mlr.press/v37/zhangc15.pdf UR - https://proceedings.mlr.press/v37/zhangc15.html AB - We study the following generalized matrix rank estimation problem: given an n-by-n matrix and a constant c > 0, estimate the number of eigenvalues that are greater than c. In the distributed setting, the matrix of interest is the sum of m matrices held by separate machines. We show that any deterministic algorithm solving this problem must communicate Ω(n^2) bits, which is order-equivalent to transmitting the whole matrix. In contrast, we propose a randomized algorithm that communicates only O(n) bits. The upper bound is matched by an Ω(n) lower bound on the randomized communication complexity. We demonstrate the practical effectiveness of the proposed algorithm with some numerical experiments. ER -
APA
Zhang, Y., Wainwright, M. & Jordan, M.. (2015). Distributed Estimation of Generalized Matrix Rank: Efficient Algorithms and Lower Bounds. Proceedings of the 32nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 37:457-465 Available from https://proceedings.mlr.press/v37/zhangc15.html.

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