Projected Tensor Power Method for Hypergraph Community Recovery

Jinxin Wang, Yuen-Man Pun, Xiaolu Wang, Peng Wang, Anthony Man-Cho So
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:36285-36307, 2023.

Abstract

This paper investigates the problem of exact community recovery in the symmetric $d$-uniform $(d \geq 2)$ hypergraph stochastic block model ($d$-HSBM). In this model, a $d$-uniform hypergraph with $n$ nodes is generated by first partitioning the $n$ nodes into $K\geq 2$ equal-sized disjoint communities and then generating hyperedges with a probability that depends on the community memberships of $d$ nodes. Despite the non-convex and discrete nature of the maximum likelihood estimation problem, we develop a simple yet efficient iterative method, called the projected tensor power method, to tackle it. As long as the initialization satisfies a partial recovery condition in the logarithmic degree regime of the problem, we show that our proposed method can exactly recover the hidden community structure down to the information-theoretic limit with high probability. Moreover, our proposed method exhibits a competitive time complexity of $\mathcal{O}(n\log^2n/\log\log n)$ when the aforementioned initialization condition is met. We also conduct numerical experiments to validate our theoretical findings.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-wang23af, title = {Projected Tensor Power Method for Hypergraph Community Recovery}, author = {Wang, Jinxin and Pun, Yuen-Man and Wang, Xiaolu and Wang, Peng and So, Anthony Man-Cho}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {36285--36307}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/wang23af/wang23af.pdf}, url = {https://proceedings.mlr.press/v202/wang23af.html}, abstract = {This paper investigates the problem of exact community recovery in the symmetric $d$-uniform $(d \geq 2)$ hypergraph stochastic block model ($d$-HSBM). In this model, a $d$-uniform hypergraph with $n$ nodes is generated by first partitioning the $n$ nodes into $K\geq 2$ equal-sized disjoint communities and then generating hyperedges with a probability that depends on the community memberships of $d$ nodes. Despite the non-convex and discrete nature of the maximum likelihood estimation problem, we develop a simple yet efficient iterative method, called the projected tensor power method, to tackle it. As long as the initialization satisfies a partial recovery condition in the logarithmic degree regime of the problem, we show that our proposed method can exactly recover the hidden community structure down to the information-theoretic limit with high probability. Moreover, our proposed method exhibits a competitive time complexity of $\mathcal{O}(n\log^2n/\log\log n)$ when the aforementioned initialization condition is met. We also conduct numerical experiments to validate our theoretical findings.} }
Endnote
%0 Conference Paper %T Projected Tensor Power Method for Hypergraph Community Recovery %A Jinxin Wang %A Yuen-Man Pun %A Xiaolu Wang %A Peng Wang %A Anthony Man-Cho So %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-wang23af %I PMLR %P 36285--36307 %U https://proceedings.mlr.press/v202/wang23af.html %V 202 %X This paper investigates the problem of exact community recovery in the symmetric $d$-uniform $(d \geq 2)$ hypergraph stochastic block model ($d$-HSBM). In this model, a $d$-uniform hypergraph with $n$ nodes is generated by first partitioning the $n$ nodes into $K\geq 2$ equal-sized disjoint communities and then generating hyperedges with a probability that depends on the community memberships of $d$ nodes. Despite the non-convex and discrete nature of the maximum likelihood estimation problem, we develop a simple yet efficient iterative method, called the projected tensor power method, to tackle it. As long as the initialization satisfies a partial recovery condition in the logarithmic degree regime of the problem, we show that our proposed method can exactly recover the hidden community structure down to the information-theoretic limit with high probability. Moreover, our proposed method exhibits a competitive time complexity of $\mathcal{O}(n\log^2n/\log\log n)$ when the aforementioned initialization condition is met. We also conduct numerical experiments to validate our theoretical findings.
APA
Wang, J., Pun, Y., Wang, X., Wang, P. & So, A.M.. (2023). Projected Tensor Power Method for Hypergraph Community Recovery. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:36285-36307 Available from https://proceedings.mlr.press/v202/wang23af.html.

Related Material