Strongly local p-norm-cut algorithms for semi-supervised learning and local graph clustering

Abstract

Graph based semi-supervised learning is the problem of learning a labeling function for the graph nodes given a few example nodes, often called seeds, usually under the assumption that the graph's edges indicate similarity of labels. This is closely related to the local graph clustering or community detection problem of finding a cluster or community of nodes around a given seed. For this problem, we propose a novel generalization of random walk, diffusion, or smooth function methods in the literature to a convex p-norm cut function. The need for our p-norm methods is that, in our study of existing methods, we find those principled methods based on eigenvector, spectral, random walk, or linear system often have difficulty capturing the correct boundary of a target label or target cluster. In contrast, 1-norm or maxflow-mincut based methods capture the boundary, but cannot grow from small seed set; hybrid procedures that use both have many hard to set parameters. In this paper, we propose a generalization of the objective function behind these methods involving p-norms. To solve the p-norm cut problem we give a strongly local algorithm -one whose runtime depends on the size of the output rather than the size of the graph. Our method can be thought as a nonlinear generalization of the Anderson-Chung-Lang push procedure to approximate a personalized PageRank vector efficiently. Our procedure is general and can solve other types of nonlinear objective functions, such as p-norm variants of Huber losses. We provide a theoretical analysis of finding planted target clusters with our method and show that the p-norm cut functions improve on the standard Cheeger inequalities for random walk and spectral methods. Finally, we demonstrate the speed and accuracy of our new method in synthetic and real world datasets. Our code is available github.com/MengLiuPurdue/SLQ.

Strongly Local Hypergraph Diffusions for Clustering and Semi-supervised Learning

“…These show that the flow-based HyperLocal method has a difficult time finding the entire cluster. Flow-based methods are known to have trouble expanding small seed sets [11,28,34] and this experiments shows that same behavior. Our strongly local hypergraph PageRank (LHPR) slightly improves on the performance of a quadratic hypergraph PageRank (QHPR) that is not strongly local.…”

“…To accomplish this, we first defined a localized directed cut graph, involving a source and sink nodes and new weighted edges. This approach is closely related to previously defined localized cut graphs for local graph clustering and semi-supervised learning [4,7,12,28,34,43], and a similar localized cut hypergraph used for flow-based hypergraph clustering [33]. The key conceptual difference is that we apply this construction directly to the reduced graph 𝐺, which by Theorem 3.3 preserves conductance of the original hypergraph H .…”

“…We first state the optimality conditions in Theorem 3.4, and then present the algorithm to solve them. The simplest way to understand this algorithm is as a generalization of the Andersen-Chung-Lang push procedure for PageRank [3], which we will call ACL as well as the more recent nonlinear push procedure [28]. Two new challenges about this new algorithm are: (1) the new algorithm operates on a directed graph, which means unlike ACL there is no single closed form update at each iteration (2) there is no sparsity regularization for auxiliary nodes, which will break the strongly local guarantees for existing analyses of the push procedure.…”

“…(See Algorithm 1 for a formalization of this strategy.) When 𝜌 < 1, we only approximately satisfy the optimality conditions; and this approximation strategy has been repeatedly and successfully used in existing local graph clustering algorithms [3,12,28].…”

“…Then we localize the computation in the reduced graph using a general strategy to build strongly local computations. This involves a particular modification often called a "localized cut" graph or hypergraph [4,11,28,33]. We then use a squared 2-norm (i.e.…”