Solving Tensor Structured Problems with Computational Tensor Algebra

TL;DR

This paper introduces a tensor algebra framework for solving multidimensional scientific problems, enabling more natural problem formulation, automated optimization, and handling of large-scale data, demonstrated on a 4D medical imaging case.

Contribution

It presents a unified tensor algebra approach that preserves data structure and enables automated optimization, outperforming traditional matrix methods in large-scale problems.

Findings

Successfully solved a 4D medical imaging problem with over 30 million unknowns.

Outperformed the best existing matrix-based approaches in efficiency.

Demonstrated applicability to large scientific datasets on standard hardware.

Abstract

Since its introduction by Gauss, Matrix Algebra has facilitated understanding of scientific problems, hiding distracting details and finding more elegant and efficient ways of computational solving. Today's largest problems, which often originate from multidimensional data, might profit from even higher levels of abstraction. We developed a framework for solving tensor structured problems with tensor algebra that unifies concepts from tensor analysis, multilinear algebra and multidimensional signal processing. In contrast to the conventional matrix approach, it allows the formulation of multidimensional problems, in a multidimensional way, preserving structure and data coherence; and the implementation of automated optimizations of solving algorithms, based on the commutativity of all tensor operations. Its ability to handle large scientific tasks is showcased by a real-world, 4D medical imaging problem, with more than 30 million unknown parameters solved on a current, inexpensive hardware. This significantly surpassed the best published matrix-based approach.