A Multi-resolution Low-rank Tensor Decomposition

TL;DR

This paper introduces a hierarchical multi-resolution low-rank tensor decomposition method that represents tensors through multiple lower-dimensional tensors, enabling efficient and structured approximation of high-order data.

Contribution

It proposes a novel multi-resolution tensor decomposition framework inspired by Tucker and PARAFAC, with an algorithm for hierarchical tensor approximation.

Findings

Preliminary simulations demonstrate the method's potential effectiveness.

The approach captures multi-scale tensor structures.

The decomposition offers a parsimonious representation of high-order tensors.

Abstract

The (efficient and parsimonious) decomposition of higher-order tensors is a fundamental problem with numerous applications in a variety of fields. Several methods have been proposed in the literature to that end, with the Tucker and PARAFAC decompositions being the most prominent ones. Inspired by the latter, in this work we propose a multi-resolution low-rank tensor decomposition to describe (approximate) a tensor in a hierarchical fashion. The central idea of the decomposition is to recast the tensor into \emph{multiple} lower-dimensional tensors to exploit the structure at different levels of resolution. The method is first explained, an alternating least squares algorithm is discussed, and preliminary simulations illustrating the potential practical relevance are provided.