Mode estimation on matrix manifolds: Convergence and robustness

Hiroaki Sasaki, Jun-Ichiro Hirayama, Takafumi Kanamori
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:8056-8079, 2022.

Abstract

Data on matrix manifolds are ubiquitous on a wide range of research fields. The key issue is estimation of the modes (i.e., maxima) of the probability density function underlying the data. For instance, local modes (i.e., local maxima) can be used for clustering, while the global mode (i.e., the global maximum) is a robust alternative to the Frechet mean. Previously, to estimate the modes, an iterative method has been proposed based on a Riemannian gradient estimator and empirically showed the superior performance in clustering (Ashizawa et al., 2017). However, it has not been theoretically investigated if the iterative method is able to capture the modes based on the gradient estimator. In this paper, we propose simple iterative methods for mode estimation on matrix manifolds based on the Euclidean metric. A key contribution is to perform theoretical analysis and establish sufficient conditions for the monotonic ascending and convergence of the proposed iterative methods. In addition, for the previous method, we prove the monotonic ascending property towards a mode. Thus, our work can be also regarded as compensating for the lack of theoretical analysis in the previous method. Furthermore, the robustness of the iterative methods is theoretically investigated in terms of the breakdown point. Finally, the proposed methods are experimentally demonstrated to work well in clustering and robust mode estimation on matrix manifolds.

Cite this Paper

BibTeX
@InProceedings{pmlr-v151-sasaki22a, title = { Mode estimation on matrix manifolds: Convergence and robustness }, author = {Sasaki, Hiroaki and Hirayama, Jun-Ichiro and Kanamori, Takafumi}, booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics}, pages = {8056--8079}, year = {2022}, editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel}, volume = {151}, series = {Proceedings of Machine Learning Research}, month = {28--30 Mar}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v151/sasaki22a/sasaki22a.pdf}, url = {https://proceedings.mlr.press/v151/sasaki22a.html}, abstract = { Data on matrix manifolds are ubiquitous on a wide range of research fields. The key issue is estimation of the modes (i.e., maxima) of the probability density function underlying the data. For instance, local modes (i.e., local maxima) can be used for clustering, while the global mode (i.e., the global maximum) is a robust alternative to the Frechet mean. Previously, to estimate the modes, an iterative method has been proposed based on a Riemannian gradient estimator and empirically showed the superior performance in clustering (Ashizawa et al., 2017). However, it has not been theoretically investigated if the iterative method is able to capture the modes based on the gradient estimator. In this paper, we propose simple iterative methods for mode estimation on matrix manifolds based on the Euclidean metric. A key contribution is to perform theoretical analysis and establish sufficient conditions for the monotonic ascending and convergence of the proposed iterative methods. In addition, for the previous method, we prove the monotonic ascending property towards a mode. Thus, our work can be also regarded as compensating for the lack of theoretical analysis in the previous method. Furthermore, the robustness of the iterative methods is theoretically investigated in terms of the breakdown point. Finally, the proposed methods are experimentally demonstrated to work well in clustering and robust mode estimation on matrix manifolds. } }
Endnote
%0 Conference Paper %T Mode estimation on matrix manifolds: Convergence and robustness %A Hiroaki Sasaki %A Jun-Ichiro Hirayama %A Takafumi Kanamori %B Proceedings of The 25th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2022 %E Gustau Camps-Valls %E Francisco J. R. Ruiz %E Isabel Valera %F pmlr-v151-sasaki22a %I PMLR %P 8056--8079 %U https://proceedings.mlr.press/v151/sasaki22a.html %V 151 %X Data on matrix manifolds are ubiquitous on a wide range of research fields. The key issue is estimation of the modes (i.e., maxima) of the probability density function underlying the data. For instance, local modes (i.e., local maxima) can be used for clustering, while the global mode (i.e., the global maximum) is a robust alternative to the Frechet mean. Previously, to estimate the modes, an iterative method has been proposed based on a Riemannian gradient estimator and empirically showed the superior performance in clustering (Ashizawa et al., 2017). However, it has not been theoretically investigated if the iterative method is able to capture the modes based on the gradient estimator. In this paper, we propose simple iterative methods for mode estimation on matrix manifolds based on the Euclidean metric. A key contribution is to perform theoretical analysis and establish sufficient conditions for the monotonic ascending and convergence of the proposed iterative methods. In addition, for the previous method, we prove the monotonic ascending property towards a mode. Thus, our work can be also regarded as compensating for the lack of theoretical analysis in the previous method. Furthermore, the robustness of the iterative methods is theoretically investigated in terms of the breakdown point. Finally, the proposed methods are experimentally demonstrated to work well in clustering and robust mode estimation on matrix manifolds.
APA
Sasaki, H., Hirayama, J. & Kanamori, T.. (2022). Mode estimation on matrix manifolds: Convergence and robustness . Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 151:8056-8079 Available from https://proceedings.mlr.press/v151/sasaki22a.html.

Related Material