Fast Tucker Rank Reduction for Non-Negative Tensors Using Mean-Field Approximation

TL;DR

This paper introduces a fast, mean-field approximation-based algorithm for non-negative tensor Tucker rank reduction, offering efficiency and competitive accuracy without gradient methods.

Contribution

It presents a novel low-rank approximation algorithm derived from mean-field approximation, eliminating the need for gradient-based optimization.

Findings

Faster than existing methods for non-negative Tucker rank reduction.

Achieves competitive or better tensor approximation quality.

Provides a theoretical condition for Tucker rank reduction via mean-field approximation.

Abstract

We present an efficient low-rank approximation algorithm for non-negative tensors. The algorithm is derived from our two findings: First, we show that rank-1 approximation for tensors can be viewed as a mean-field approximation by treating each tensor as a probability distribution. Second, we theoretically provide a sufficient condition for distribution parameters to reduce Tucker ranks of tensors; interestingly, this sufficient condition can be achieved by iterative application of the mean-field approximation. Since the mean-field approximation is always given as a closed formula, our findings lead to a fast low-rank approximation algorithm without using a gradient method. We empirically demonstrate that our algorithm is faster than the existing non-negative Tucker rank reduction methods and achieves competitive or better approximation of given tensors.