Uniform Hypergraph Partitioning: Provable Tensor Methods and Sampling Techniques

Graph partitioning plays a central role in machine learning, and the development of graph partitioning algorithms is still an active area of research. The immense demand for such algorithms arises due to the abundance of applications that involve pairwise interactions or similarities among entities. Recent studies in computer vision and databases systems have emphasized on the necessity of considering multi-way interactions, and has led to the study of a more general problem in the form of hypergraph partitioning. This paper focuses on the problem of partitioning uniform hypergraphs, which arises in computer vision applications such as subspace clustering, motion segmentation etc. We show that uniform hypergraph partitioning is equivalent to a tensor trace maximization problem, and hence, a tensor based method is a natural answer to this problem. We also propose a tensor spectral method that extends the widely known spectral clustering algorithm to the case of uniform hypergraphs. While the theoretical guarantees of spectral clustering have been extensively studied, very little is known about the statistical properties of tensor based methods. To this end, we prove the consistency of the proposed algorithm under a planted partition model. The computational complexity of tensorial approaches has resulted in the use of various tensor sampling strategies. We present the first theoretical study on the effect of sampling in tensor based hypergraph partitioning. Our result justifies the empirical success of iterative sampling techniques often used in practice. We also present an iteratively sampled variant of the proposed algorithm for the purpose of subspace clustering, and demonstrate the performance of this method on a benchmark problem.