Uniform Hypergraph Partitioning: Provable Tensor Methods and Sampling Techniques

TL;DR

This paper develops provable, efficient algorithms for partitioning weighted uniform hypergraphs, addressing computational challenges and justifying sampling techniques through theoretical analysis and empirical comparisons.

Contribution

It introduces new algorithms for hypergraph partitioning that are both provably correct and computationally efficient, especially for sparse, weighted hypergraphs in computer vision.

Findings

Sampling strategies are empirically justified by theoretical analysis.

Proposed algorithms outperform existing methods in efficiency and accuracy.

The work extends previous results to weighted hypergraphs in practical applications.

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.