Strongly Local Hypergraph Diffusions for Clustering and Semi-supervised Learning

Abstract

Hypergraph-based machine learning methods are now widely recognized as important for modeling and using higher-order and multiway relationships between data objects. Local hypergraph clustering and semi-supervised learning specifically involve finding a well-connected set of nodes near a given set of labeled vertices. Although many methods for local graph clustering exist, there are relatively few for localized clustering in hypergraphs. Moreover, those that exist often lack flexibility to model a general class of hypergraph cut functions or cannot scale to large problems. To tackle these issues, this paper proposes a new diffusion-based hypergraph clustering algorithm that solves a quadratic hypergraph cut based objective akin to a hypergraph analog of Andersen-Chung-Lang personalized PageRank clustering for graphs. We prove that, for graphs with fixed maximum hyperedge size, this method is strongly local, meaning that its runtime only depends on the size of the output instead of the size of the hypergraph and is highly scalable. Moreover, our method enables us to compute with a wide variety of cardinality-based hypergraph cut functions. We also prove that the clusters found by solving the new objective function satisfy a Cheeger-like quality guarantee. We demonstrate that on large real-world hypergraphs our new method finds better clusters and runs much faster than existing approaches. Specifically, it runs in a few seconds for hypergraphs with a few million hyperedges compared with minutes for a flow-based technique. We furthermore show that our framework is general enough that can also be used to solve other p-norm based cut objectives on hypergraphs. CCS CONCEPTS• Mathematics of computing → Hypergraphs; • Theory of computation → Semi-supervised learning; • Computing methodologies → Cluster analysis.

Approximate Decomposable Submodular Function Minimization for Cardinality-Based Components

“…In addition to its strong theoretical guarantees, SPARSECARD is very practical and leads to substantial improvements in benchmark image segmentation problems and hypergraph clustering tasks. We focus on DSFM problems that simultaneously include component functions of large and small support, which are common in computer vision and hypergraph clustering applications [40,11,43,34,39]. We ran experiments on a laptop with a 2.2 GHz Intel Core i7 processor and 8GB of RAM.…”

“…Hypergraph local clustering. Graph reduction techniques have been frequently and successfully used as subroutines for hypergraph local clustering and semi-supervised learning methods [34,44,30,48]. Replacing exact reductions with our approximate reductions can lead to significant runtime improvements without sacrificing on accuracy, and opens the door to running local clustering algorithms on problems where exact graph reduction would be infeasible.…”

“…where for each e ∈ E, f e is a submodular function supported on a subset e ⊆ V . Functions of this form often arise as energy functions in computer vision [24,25,15], and generalized cut functions for hypergraph clustering and learning [14,30,32,34,43,44], among other applications.…”

“…In practice, cardinality-based decomposable submodular functions frequently arise as higher-order energy functions in computer vision [24] and set cover functions [41]. The most widely used generalized hypergraph cut functions are also cardinality based [32,34,43,44,1]. Even previous research on algorithms for the more general DSFM problem tends to focus on cardinality-based examples in experimental results [11,41,21,31].…”

Higher-order Network Analysis Takes Off, Fueled by Classical Ideas and New Data

“…Using a spacey random walk on that same higher-order Markov chain, however, yields a tensor eigenvector [10,21]. There are related notions of PageRank on hypergraphs [36,39] and motif-derived hypergraphs [55]. These use the ability of hypergraph edges to be associated with functions that model complex cuts [37,53].…”