Research Track Paper KDD '20, August 23–27, 2020, Virtual Event, USA Minimizing Localized Ratio Cut Objectives in Hypergraphs Nate Veldt Austin R. Benson Jon Kleinberg Cornell University Cornell University Cornell University Center for Applied Mathematics Department of Computer Science Department of Computer Science [email protected] [email protected] [email protected] ABSTRACT to identify communities [13], predict class labels in machine learn- Hypergraphs are a useful abstraction for modeling multiway rela- ing applications [7], and segment images [18]. Standard models tionships in data, and hypergraph clustering is the task of detecting for such clusters are ratio cut objectives, which measure the ratio groups of closely related nodes in such data. Graph clustering has between the number of edges leaving a cluster (the cut) and some been studied extensively, and there are numerous methods for de- notion of the cluster’s size (e.g., the number of edges or nodes in the tecting small, localized clusters without having to explore an entire cluster); common ratio cut objectives include conductance, sparsest input graph. However, there are only a few specialized approaches cut, and normalized cut. Ratio cut objectives are intimately related for localized clustering in hypergraphs. Here we present a frame- to spectral clustering techniques, with the latter providing approxi- work for local hypergraph clustering based on minimizing localized mation guarantees for ratio cut objectives (such as conductance) ratio cut objectives. Our framework takes an input set of reference via so-called Cheeger inequalities [11]. In some cases, these ratio nodes in a hypergraph and solves a sequence of hypergraph mini- cut objectives are optimized over an entire graph to solve a global mum s-t cut problems in order to identify a nearby well-connected clustering or classification task [18]. In other situations, the goal is cluster of nodes that overlaps substantially with the input set. to find sets of nodes that have a small ratio cut and are localized to Our methods extend graph-based techniques but are significantly a certain region of a large graph [3, 4, 27, 34]. more general and have new output quality guarantees. First, our Recently, there has been a surge of hypergraph methods for ma- methods can minimize new generalized notions of hypergraph chine learning and data mining [1, 2, 5, 10, 23, 25, 37, 39], as hyper- cuts, which depend on specific configurations of nodes within each graphs can better model multiway relationships in data. Common hyperedge, rather than just on the number of cut hyperedges. Sec- examples of multiway relationships include academic researchers ond, our framework has several attractive theoretical properties in co-authoring papers, retail items co-purchased by shoppers, and terms of output cluster quality. Most importantly, our algorithm is sets of products or services reviewed by the same person. Due to strongly-local, meaning that its runtime depends only on the size broader modeling ability, there are many hypergraph generaliza- of the input set, and does not need to explore the entire hypergraph tions of graph-cut objectives, including hypergraph variants of ratio to find good local clusters. We use our methodology to effectively cut objectives like conductance and normalized cut [6, 8, 9, 23, 39]. identify clusters in hypergraphs of real-world data with millions of Nevertheless, there are numerous challenges in extending graph- nodes, millions of hyperedges, and large average hyperedge size cut techniques to the hypergraph setting, and current methods with runtimes ranging between a few seconds and a few minutes. for hypergraph-based learning are much less developed than their graph-based counterparts. One major challenge in generalizing ACM Reference Format: graph cut methods is that the concept of a cut hyperedge — how Nate Veldt, Austin R. Benson, and Jon Kleinberg. 2020. Minimizing Localized to define it and how to penalize it — is more nuanced thanthe Ratio Cut Objectives in Hypergraphs. In Proceedings of the 26th ACM SIGKDD concept of a cut edge. While there is only one way to separate the Conference on Knowledge Discovery and Data Mining (KDD ’20), August 23–27, 2020, Virtual Event, CA, USA. ACM, New York, NY, USA, 11 pages. endpoints of an edge into two clusters, there are several ways to https://doi.org/10.1145/3394486.3403222 split up a set of three or more nodes in a hypergedge. Many objective functions model cuts with an all-or-nothing penalty function, which assigns the same penalty to any way of splitting up the nodes of the 1 INTRODUCTION hyperedge (and a penalty of zero if all nodes in the hyperedge are Graphs are a common mathematical abstraction for modeling pair- placed together) [16, 17, 21] (Fig. 1a). However, a common practical wise interactions between objects in a dataset. A standard task in heuristic is a clique expansion, which replaces each hyperedge with graph-based data analysis is to identify well-connected clusters of a weighted clique in a graph [6, 16, 23, 25, 39, 40] (Fig. 1b). The nodes, which share more edges with each other than the rest of advantage is that graph methods can be directly applied, but this the graph [30]. For example, detecting clusters in a graph is used heuristic actually penalizes cut hyperedges differently than the all- or-nothing model. Another downside is that for hypergraph with Permission to make digital or hard copies of all or part of this work for personal or large hyperedges, clique expansion produces a very dense graph. classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation Solving an all-or-nothing cut or applying clique expansion are on the first page. Copyrights for components of this work owned by others than the only two specific models for higher-order relationships. And ifone author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or uses a ratio cut model for clusters in a hypergraph, how to penalize republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. a cut hyperedge may depend on the application. Along these lines, KDD ’20, August 23–27, 2020, Virtual Event, CA, USA inhomogeneous hypergraphs model every possible way to split up a © 2020 Copyright held by the owner/author(s). Publication rights licensed to ACM. hyperedge to assign different penalties [23]; however, these more ACM ISBN 978-1-4503-7998-4/20/08...$15.00 https://doi.org/10.1145/3394486.3403222 sophisticated models are still approximated with weighted clique 1708 Research Track Paper KDD '20, August 23–27, 2020, Virtual Event, USA A major feature of our methods is that they run in strongly-local penalty = w (A) penalty 0,w <latexit sha1_base64="UEJT5+8D0X9OI8ullh/5OE+XvqY=">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</latexit> e time <latexit sha1_base64="WR9A8BSWHwq2U8N0UqbYyS5Y5J4=">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</latexit> penalty = c( A e A ) , meaning that the runtime is dependent only on the size of e <latexit sha1_base64="IMax4W/PS4BbwAgH7hb17MtAuNU=">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</latexit> 2 { } | |·| \ | A<latexit sha1_base64="Y1jZPZQ/1y5QPUmPr0XFTLTeAiM=">AAAChHicbVFNb9NAEF0bKMV8NMCRy4iookhVZKcgEFJRAxeOBZG2UjZE6804WXW9tnbHhMjxL+FfcePfsEl9gISRVnp67432zUxaauUojn8H4a3bd/bu7t+L7j94+Oig8/jJhSsqK3EoC13Yq1Q41MrgkBRpvCotijzVeJlef1zrl9/ROlWYr7QscZyLmVGZkoI8Nen8POSEP6gu0QhNywZOYTVYcTktCFbIUyGvnRZuDoNVtG3lygCv42NYTBB4s6OfwoZIs3rRTPBo8JLz1tNS8A7633yrVbM5CWuLBfBc0DxN6y9NNJh0unEv3hTsgqQFXdbW+aTzi08LWeVoSPrQbpTEJY1rYUlJjU3EK4eln0jMcOShETm6cb1ZYgOHnplCVlj/DMGG/bujFrlzyzz1znVGt62tyf9po4qyt+NambIiNPLmo6zSQAWsLwJTZVGSXnogpFU+K8i5sEKSv1vkl5Bsj7wLLvq95KTX//yqe/ahXcc+e8aesyOWsDfsjH1i52zIZBAEL4I4SMK98Dg8CV/fWMOg7XnK/qnw/R9t7MII</latexit> the input set, rather than the entire hypergraph. Therefore, we can A<latexit sha1_base64="Y1jZPZQ/1y5QPUmPr0XFTLTeAiM=">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</latexit>