Spectral Partitioning of Large and Sparse Tensors using Low-Rank Tensor Approximation

TL;DR

This paper extends spectral graph partitioning techniques to large, sparse tensors by developing a low-rank tensor approximation method that reveals underlying reducibility structures and aids in data partitioning.

Contribution

It introduces a novel spectral partitioning approach for tensors based on low-rank approximations, generalizing spectral graph methods to higher dimensions.

Findings

The method accurately detects reducibility structures in tensors.

It performs well on synthetic and real application data.

The approach is robust to noise and sparsity.

Abstract

The problem of partitioning a large and sparse tensor is considered, where the tensor consists of a sequence of adjacency matrices. Theory is developed that is a generalization of spectral graph partitioning. A best rank-$(2,2,\lambda)$ approximation is computed for $\lambda=1,2,3$, and the partitioning is computed from the orthogonal matrices and the core tensor of the approximation. It is shown that if the tensor has a certain reducibility structure, then the solution of the best approximation problem exhibits the reducibility structure of the tensor. Further, if the tensor is close to being reducible, then still the solution of the exhibits the structure of the tensor. Numerical examples with synthetic data corroborate the theoretical results. Experiments with tensors from applications show that the method can be used to extract relevant information from large, sparse, and noisy data.