Hypergraph p-Laplacian: A Differential Geometry View

The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph pLaplacian. Unlike the existing two-node graph Laplacians, this generalization makes it possible to analyze hypergraphs, where the edges are allowed to connect any number of nodes. Moreover, we propose a semi-supervised learning method based on the proposed hypergraph p-Laplacian, and formalize them as the analogue to the Dirichlet problem, which often appears in physics. We further explore theoretical connections to normalized hypergraph cut on a hypergraph, and propose normalized cut corresponding to hypergraph p-Laplacian. The proposed p-Laplacian is shown to outperform standard hypergraph Laplacians in the experiment on a hypergraph semisupervised learning and normalized cut setting. Introduction Graphs are a standard way to represent pairwise relationship data on both regular and irregular domains. One of the most important operators characterizing a graph is the graph Laplacian, which can be explained in several ways. For the example of spectral clustering (von Luxburg 2007), we consider normalized graph cut (Shi and Malik 1997; Yu and Shi 2003), random walks (Meila and Shi 2001; Grady 2006), and analogues to differential geometry of graphs (Branin 1966; Grady and Schwartz 2003; Zhou and Schölkopf 2006; Bougleux, Elmoataz, and Melkemi 2007). Hypergraphs are a natural generalization of graphs, where the edges are allowed to connect more than two nodes (Berge 1984). The data representation with a hypergraph is used in a variety of applications (Huang, Liu, and Metaxas 2009; Liu, Latecki, and Yan 2010; Klamt, Haus, and Theis 2009; Tan et al. 2014). This natural generalization of graphs motivates us to consider a natural generalization of Laplacian to hypergraphs, which can be applied to hypergraph clustering problems. However, there is no straightforward approach to generalize the graph Laplacian to a hypergraph Laplacian. One way is to model a hypergraph as a tensor, for which we can define Laplacian (Cooper and Dutle 2012; Hu and Qi Copyright c © 2018, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. 2015) and construct hypergraph cut algorithms (Bulò and Pelillo 2009; Ghoshdastidar and Dukkipati 2014). However, this requires the hypergraph to obey a strict condition of a k-uniform hypergraph, where each edge connects exactly k nodes. The second approach is to construct a weighted graph, which can deal with arbitrary hypergraphs. Rodriguez’s approach defines Laplacian of arbitrary hypergraph as an adjacency matrix of weighted graph (Rodriguez 2002). Zhou’s approach defines a hypergraph from the normalized cut approach, and outperforms Rodriguez’s Laplacian on a clustering problem (Zhou, Huang, and Schölkopf 2006). However, Rodriguez’s Laplacian does not consider how many nodes are connected by each edge, and Zhou’s Laplacian is not consistent with the graph Laplacian. Although all of previous studies consider the analogue to graph Laplacian, none of them considers the analogue to the Laplacian from differential geometry. This allows us to further extend to more general hypergraph p-Laplacian, which is not extensively studied unlike in the case of graph p-Laplacian (Bühler and Hein 2009; Zhou and Schölkopf 2006). In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph p-Laplacian, which is consistent with the graph Laplacian. We define gradient of the function over hypergraph, and induce the divergence and Laplacian as formulated in differential geometry. Taking advantage of this formulation, we extend our hypergraph Laplacian to a hypergraph p-Laplacian, which allows us to better capture hypergraph characteristics. We also propose a semi-supervised machine learning method based upon this p-Laplacian. Our experiment on hypergraph semi-supervised clustering problem shows that our hypergraph p-Laplacian outperforms the current hypergraph Laplacians. The versatility of differential geometry allows us to introduce several rigorous interpretations of our hypergraph Laplacian. A normalized cut formulation is shown to yield the proposed hypergraph Laplacian in the same manner as in standard graphs. We further propose a normalized cut corresponding to our p-Laplacian, which shows better performance than the ones corresponding to current Laplacians in the experiments. We also explore the physical interpretation of hypergraph Laplacian, by considering the analogue to the continuous p-Dirichlet problem, which is widely used in Physics. All proofs are in Appendix Section. ar X iv :1 71 1. 08 17 1v 1 [ st at .M L ] 2 2 N ov 2 01 7 Differential Geometry on Hypergraphs Preliminary Definition of Hypergraph In this section, we review standard definitions and notations from hypergraph theory. We refer to the literature (Berge 1984) for a more comprehensive study. A hypergraph G is a pair (V,E), where E ⊂ ∪ | k=1∪{v1,...,vk}⊂V {[vσ(1), . . . , vσ(k)] | σ ∈ Sk}, and Sk denotes the set of permutations σ on {1, . . . , k}. An element of V is called a vertex or node, and an element of E is referred to as an edge or hyperedge of the hypergraph. A hypergraph is connected if the intersection graph of the edges is connected. In what follows, we assume that the hypergraph G is connected. A hypergraph is undirected when the set of edges are symmetric, and we denote a set of undirected edges as Eun = E/S, where S = ∪ | k=1Sk. In other words, edges [v1, v2, . . . , vk] ∈ E and [vσ(1), vσ(2), . . . , vσ(k)] ∈ E are not distinguished inEun for any σ ∈ Sk, where k is the number of nodes in the edge. A hypergraph is weighted when it is associated with a function w : E → R. For an undirected hypergraph it holds that w([v1, v2, . . . , vk])=w([vσ(1), vσ(2), . . . , vσ(k)]). We define the degree of a node v ∈ V as d(v) = ∑ e∈E:v∈e w(e), while the degree of an edge e ∈ E is defined as δ(e) = |e|. To simplify the notation we write δe instead of δ(e). We defineH(V ) as a Hilbert space of real-valued functions endowed with the usual inner product