Hypergraph Spectral Clustering in the Weighted Stochastic Block Model

TL;DR

This paper introduces two algorithms for hypergraph spectral clustering in the weighted stochastic block model, providing performance guarantees and demonstrating optimality and practical applications in subspace clustering.

Contribution

The paper develops and analyzes two hypergraph spectral clustering algorithms with provable performance guarantees under the weighted stochastic block model, improving state-of-the-art results.

Findings

HSC outperforms random guessing when total edge weight is Ω(n).

HSC recovers the hidden partition with vanishing error when total weight is ω(n).

HSCLR exactly recovers the hidden partition when total weight is on the order of n log n.

Abstract

Spectral clustering is a celebrated algorithm that partitions objects based on pairwise similarity information. While this approach has been successfully applied to a variety of domains, it comes with limitations. The reason is that there are many other applications in which only \emph{multi}-way similarity measures are available. This motivates us to explore the multi-way measurement setting. In this work, we develop two algorithms intended for such setting: Hypergraph Spectral Clustering (HSC) and Hypergraph Spectral Clustering with Local Refinement (HSCLR). Our main contribution lies in performance analysis of the poly-time algorithms under a random hypergraph model, which we name the weighted stochastic block model, in which objects and multi-way measures are modeled as nodes and weights of hyperedges, respectively. Denoting by $n$ the number of nodes, our analysis reveals the following: (1) HSC outputs a partition which is better than a random guess if the sum of edge weights (to be explained later) is $\Omega(n)$; (2) HSC outputs a partition which coincides with the hidden partition except for a vanishing fraction of nodes if the sum of edge weights is $\omega(n)$; and (3) HSCLR exactly recovers the hidden partition if the sum of edge weights is on the order of $n \log n$. Our results improve upon the state of the arts recently established under the model and they firstly settle the order-wise optimal results for the binary edge weight case. Moreover, we show that our results lead to efficient sketching algorithms for subspace clustering, a computer vision application. Lastly, we show that HSCLR achieves the information-theoretic limits for a special yet practically relevant model, thereby showing no computational barrier for the case.