Identification of Matrix Joint Block Diagonalization

Yunfeng Cai, Ping Li
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:1495-1503, 2021.

Abstract

Given a set $\mathcal{C}=\{C_i\}_{i=1}^m$ of square matrices, the matrix blind joint block diagonalization problem (BJBDP) is to find a full column rank matrix $A$ such that $C_i=A\Sigma_iA^{\T}$ for all $i$, where $\Sigma_i$’s are all block diagonal matrices with as many diagonal blocks as possible. The BJBDP plays an important role in independent subspace analysis. This paper considers the identification problem for BJBDP, that is, under what conditions and by what means, we can identify the diagonalizer $A$ and the block diagonal structure of $\Sigma_i$, especially when there is noise in $C_i$’s. In this paper, we propose a “bi-block diagonalization” method to solve BJBDP, and establish sufficient conditions for when the method is able to accomplish the task. Numerical simulations validate our theoretical results. To the best of the authors’ knowledge, current numerical methods for BJBDP have no theoretical guarantees for the identification of the exact solution, whereas our method does.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-cai21a, title = { Identification of Matrix Joint Block Diagonalization }, author = {Cai, Yunfeng and Li, Ping}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {1495--1503}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/cai21a/cai21a.pdf}, url = {https://proceedings.mlr.press/v130/cai21a.html}, abstract = { Given a set $\mathcal{C}=\{C_i\}_{i=1}^m$ of square matrices, the matrix blind joint block diagonalization problem (BJBDP) is to find a full column rank matrix $A$ such that $C_i=A\Sigma_iA^{\T}$ for all $i$, where $\Sigma_i$’s are all block diagonal matrices with as many diagonal blocks as possible. The BJBDP plays an important role in independent subspace analysis. This paper considers the identification problem for BJBDP, that is, under what conditions and by what means, we can identify the diagonalizer $A$ and the block diagonal structure of $\Sigma_i$, especially when there is noise in $C_i$’s. In this paper, we propose a “bi-block diagonalization” method to solve BJBDP, and establish sufficient conditions for when the method is able to accomplish the task. Numerical simulations validate our theoretical results. To the best of the authors’ knowledge, current numerical methods for BJBDP have no theoretical guarantees for the identification of the exact solution, whereas our method does. } }
Endnote
%0 Conference Paper %T Identification of Matrix Joint Block Diagonalization %A Yunfeng Cai %A Ping Li %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-cai21a %I PMLR %P 1495--1503 %U https://proceedings.mlr.press/v130/cai21a.html %V 130 %X Given a set $\mathcal{C}=\{C_i\}_{i=1}^m$ of square matrices, the matrix blind joint block diagonalization problem (BJBDP) is to find a full column rank matrix $A$ such that $C_i=A\Sigma_iA^{\T}$ for all $i$, where $\Sigma_i$’s are all block diagonal matrices with as many diagonal blocks as possible. The BJBDP plays an important role in independent subspace analysis. This paper considers the identification problem for BJBDP, that is, under what conditions and by what means, we can identify the diagonalizer $A$ and the block diagonal structure of $\Sigma_i$, especially when there is noise in $C_i$’s. In this paper, we propose a “bi-block diagonalization” method to solve BJBDP, and establish sufficient conditions for when the method is able to accomplish the task. Numerical simulations validate our theoretical results. To the best of the authors’ knowledge, current numerical methods for BJBDP have no theoretical guarantees for the identification of the exact solution, whereas our method does.
APA
Cai, Y. & Li, P.. (2021). Identification of Matrix Joint Block Diagonalization . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:1495-1503 Available from https://proceedings.mlr.press/v130/cai21a.html.

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