Low-rank nonnegative tensor approximation via alternating projections and sketching

TL;DR

This paper introduces a novel method for nonnegative low-rank tensor approximation using alternating projections and sketching, improving efficiency and negative element decay in Tucker and tensor train formats.

Contribution

It presents the first study of nonnegative tensor train approximation and enhances Tucker approximation with faster algorithms and better negative element control.

Findings

Accelerated alternating projections effectively reduce negative elements.

The Tucker approximation method outperforms previous approaches in complexity.

Tensor train approximation is introduced as a new approach in this context.

Abstract

We show how to construct nonnegative low-rank approximations of nonnegative tensors in Tucker and tensor train formats. We use alternating projections between the nonnegative orthant and the set of low-rank tensors, using STHOSVD and TTSVD algorithms, respectively, and further accelerate the alternating projections using randomized sketching. The numerical experiments on both synthetic data and hyperspectral images show the decay of the negative elements and that the error of the resulting approximation is close to the initial error obtained with STHOSVD and TTSVD. The proposed method for the Tucker case is superior to the previous ones in terms of computational complexity and decay of negative elements. The tensor train case, to the best of our knowledge, has not been studied before.