Many-body Approximation for Non-negative Tensors

TL;DR

This paper introduces a novel many-body approximation method for non-negative tensors that models interactions as probability distributions, enabling global optimization and providing insights into tensor structure beyond traditional low-rank approaches.

Contribution

The paper proposes an energy-based many-body approximation approach that avoids low-rank assumptions and allows intuitive tuning of mode interactions for tensor decomposition.

Findings

Effective tensor completion demonstrated

Global optimization via KL divergence achieved

Revealed relationship between many-body and low-rank approximations

Abstract

We present an alternative approach to decompose non-negative tensors, called many-body approximation. Traditional decomposition methods assume low-rankness in the representation, resulting in difficulties in global optimization and target rank selection. We avoid these problems by energy-based modeling of tensors, where a tensor and its mode correspond to a probability distribution and a random variable, respectively. Our model can be globally optimized in terms of the KL divergence minimization by taking the interaction between variables (that is, modes), into account that can be tuned more intuitively than ranks. Furthermore, we visualize interactions between modes as tensor networks and reveal a nontrivial relationship between many-body approximation and low-rank approximation. We demonstrate the effectiveness of our approach in tensor completion and approximation.