Scalable Multiple Kernel Clustering: Learning Clustering Structure from Expectation

Weixuan Liang, En Zhu, Shengju Yu, Huiying Xu, Xinzhong Zhu, Xinwang Liu
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:29700-29719, 2024.

Abstract

In this paper, we derive an upper bound of the difference between a kernel matrix and its expectation under a mild assumption. Specifically, we assume that the true distribution of the training data is an unknown isotropic Gaussian distribution. When the kernel function is a Gaussian kernel, and the mean of each cluster is sufficiently separated, we find that the expectation of a kernel matrix can be close to a rank-$k$ matrix, where $k$ is the cluster number. Moreover, we prove that the normalized kernel matrix of the training set deviates (w.r.t. Frobenius norm) from its expectation in the order of $\widetilde{\mathcal{O}}(1/\sqrt{d})$, where $d$ is the dimension of samples. Based on the above theoretical results, we propose a novel multiple kernel clustering framework which attempts to learn the information of the expectation kernel matrices. First, we aim to minimize the distance between each base kernel and a rank-$k$ matrix, which is a proxy of the expectation kernel. Then, we fuse these rank-$k$ matrices into a consensus rank-$k$ matrix to find the clustering structure. Using an anchor-based method, the proposed framework is flexible with the sizes of input kernel matrices and able to handle large-scale datasets. We also provide the approximation guarantee by deriving two non-asymptotic bounds for the consensus kernel and clustering indicator matrices. Finally, we conduct extensive experiments to verify the clustering performance of the proposed method and the correctness of the proposed theoretical results.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-liang24g, title = {Scalable Multiple Kernel Clustering: Learning Clustering Structure from Expectation}, author = {Liang, Weixuan and Zhu, En and Yu, Shengju and Xu, Huiying and Zhu, Xinzhong and Liu, Xinwang}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {29700--29719}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/liang24g/liang24g.pdf}, url = {https://proceedings.mlr.press/v235/liang24g.html}, abstract = {In this paper, we derive an upper bound of the difference between a kernel matrix and its expectation under a mild assumption. Specifically, we assume that the true distribution of the training data is an unknown isotropic Gaussian distribution. When the kernel function is a Gaussian kernel, and the mean of each cluster is sufficiently separated, we find that the expectation of a kernel matrix can be close to a rank-$k$ matrix, where $k$ is the cluster number. Moreover, we prove that the normalized kernel matrix of the training set deviates (w.r.t. Frobenius norm) from its expectation in the order of $\widetilde{\mathcal{O}}(1/\sqrt{d})$, where $d$ is the dimension of samples. Based on the above theoretical results, we propose a novel multiple kernel clustering framework which attempts to learn the information of the expectation kernel matrices. First, we aim to minimize the distance between each base kernel and a rank-$k$ matrix, which is a proxy of the expectation kernel. Then, we fuse these rank-$k$ matrices into a consensus rank-$k$ matrix to find the clustering structure. Using an anchor-based method, the proposed framework is flexible with the sizes of input kernel matrices and able to handle large-scale datasets. We also provide the approximation guarantee by deriving two non-asymptotic bounds for the consensus kernel and clustering indicator matrices. Finally, we conduct extensive experiments to verify the clustering performance of the proposed method and the correctness of the proposed theoretical results.} }
Endnote
%0 Conference Paper %T Scalable Multiple Kernel Clustering: Learning Clustering Structure from Expectation %A Weixuan Liang %A En Zhu %A Shengju Yu %A Huiying Xu %A Xinzhong Zhu %A Xinwang Liu %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-liang24g %I PMLR %P 29700--29719 %U https://proceedings.mlr.press/v235/liang24g.html %V 235 %X In this paper, we derive an upper bound of the difference between a kernel matrix and its expectation under a mild assumption. Specifically, we assume that the true distribution of the training data is an unknown isotropic Gaussian distribution. When the kernel function is a Gaussian kernel, and the mean of each cluster is sufficiently separated, we find that the expectation of a kernel matrix can be close to a rank-$k$ matrix, where $k$ is the cluster number. Moreover, we prove that the normalized kernel matrix of the training set deviates (w.r.t. Frobenius norm) from its expectation in the order of $\widetilde{\mathcal{O}}(1/\sqrt{d})$, where $d$ is the dimension of samples. Based on the above theoretical results, we propose a novel multiple kernel clustering framework which attempts to learn the information of the expectation kernel matrices. First, we aim to minimize the distance between each base kernel and a rank-$k$ matrix, which is a proxy of the expectation kernel. Then, we fuse these rank-$k$ matrices into a consensus rank-$k$ matrix to find the clustering structure. Using an anchor-based method, the proposed framework is flexible with the sizes of input kernel matrices and able to handle large-scale datasets. We also provide the approximation guarantee by deriving two non-asymptotic bounds for the consensus kernel and clustering indicator matrices. Finally, we conduct extensive experiments to verify the clustering performance of the proposed method and the correctness of the proposed theoretical results.
APA
Liang, W., Zhu, E., Yu, S., Xu, H., Zhu, X. & Liu, X.. (2024). Scalable Multiple Kernel Clustering: Learning Clustering Structure from Expectation. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:29700-29719 Available from https://proceedings.mlr.press/v235/liang24g.html.

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