Graph Regularized Tensor Train Decomposition

TL;DR

This paper introduces a graph regularized tensor train decomposition that captures low-dimensional tensor structures while preserving local geometric relationships, improving unsupervised learning on high-dimensional tensor data.

Contribution

It proposes a novel graph regularized tensor train model formulated as a nonconvex optimization problem on the Stiefel manifold, with an efficient solution algorithm.

Findings

Outperforms existing tensor dimensionality reduction methods

Preserves local geometric relationships in tensor data

Effective for unsupervised learning applications

Abstract

With the advances in data acquisition technology, tensor objects are collected in a variety of applications including multimedia, medical and hyperspectral imaging. As the dimensionality of tensor objects is usually very high, dimensionality reduction is an important problem. Most of the current tensor dimensionality reduction methods rely on finding low-rank linear representations using different generative models. However, it is well-known that high-dimensional data often reside in a low-dimensional manifold. Therefore, it is important to find a compact representation, which uncovers the low dimensional tensor structure while respecting the intrinsic geometry. In this paper, we propose a graph regularized tensor train (GRTT) decomposition that learns a low-rank tensor train model that preserves the local relationships between tensor samples. The proposed method is formulated as a nonconvex optimization problem on the Stiefel manifold and an efficient algorithm is proposed to solve it. The proposed method is compared to existing tensor based dimensionality reduction methods as well as tensor manifold embedding methods for unsupervised learning applications.