Solving Tensor Structured Problems with Computational Tensor Algebra
For researchers working with large-scale multidimensional data, this framework offers a new paradigm that preserves data structure and enables automated optimizations, surpassing existing matrix methods.
The paper introduces a computational tensor algebra framework for solving tensor-structured problems, demonstrating its effectiveness on a 4D medical imaging problem with over 30 million unknowns, outperforming the best published matrix-based approach on inexpensive hardware.
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.