Uniform Hypergraph Partitioning: Provable Tensor Methods and Sampling Techniques

Abstract

In a series of recent works, we have generalised the consistency results in the stochastic block model literature to the case of uniform and non-uniform hypergraphs. The present paper continues the same line of study, where we focus on partitioning weighted uniform hypergraphs---a problem often encountered in computer vision. This work is motivated by two issues that arise when a hypergraph partitioning approach is used to tackle computer vision problems: (i) The uniform hypergraphs constructed for higher-order learning contain all edges, but most have negligible weights. Thus, the adjacency tensor is nearly sparse, and yet, not binary. (ii) A more serious concern is that standard partitioning algorithms need to compute all edge weights, which is computationally expensive for hypergraphs. This is usually resolved in practice by merging the clustering algorithm with a tensor sampling strategy---an approach that is yet to be analysed rigorously. We build on our earlier work on partitioning dense unweighted uniform hypergraphs (Ghoshdastidar and Dukkipati, ICML, 2015), and address the aforementioned issues by proposing provable and efficient partitioning algorithms. Our analysis justifies the empirical success of practical sampling techniques. We also complement our theoretical findings by elaborate empirical comparison of various hypergraph partitioning schemes.

To appear in Journal of Machine Learning Research (vol 18)

References in corpus (8)

• Random hypergraphs and their applications
• Provable Tensor Factorization with Missing Data
• Alignment and integration of complex networks by hypergraph-based spectral clustering
• Consistency of Spectral Hypergraph Partitioning under Planted Partition Model
• A statistical model for tensor PCA
• Greedy Subspace Clustering
• Achieving Optimal Misclassification Proportion in Stochastic Block Model
• A New Sampling Technique for Tensors

Cited by in corpus (6)

• Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond
• Hypergraph Spectral Clustering in the Weighted Stochastic Block Model
• Hypergraph Modeling via Spectral Embedding Connection: Hypergraph Cut, Weighted Kernel $k$-means, and Heat Kernel
• Multiway Spherical Clustering via Degree-Corrected Tensor Block Models
• Partial recovery and weak consistency in the non-uniform hypergraph Stochastic Block Model
• A Family of Pairwise Multi-Marginal Optimal Transports that Define a Generalized Metric