Disjoint and nearly disjoint sums of matrix multiplication tensors and their centroids
2026-08-27 • Computational Complexity
Computational Complexity
AI summaryⓘ
The authors study centroids, which are important properties of mathematical objects called tensors. They find new examples of tensors with very large centroids, disproving a previous belief that such tensors might not exist. Using a new geometric method involving centroids, they show how to break down tensors in a simpler way and prove some tensors have the smallest possible complexity. They also create special symmetric tensors that behave in a complex, 'wild' manner. Their approach improves known methods related to matrix multiplication, potentially leading to faster calculations.
tensorcentroidborder ranktensor decompositionsymmetric tensormatrix multiplication exponentStrassen's laser methodgeometric methodswild tensors
Authors
Martin Kassabov, J. M. Landsberg, Victor Souza, Philip Speegle
Abstract
This paper addresses centroids, which are fundamental invariants of tensors. Our main results are as follows: (i) The construction of explicit tensors with very large centroids, whereas previously it had been conjectured that none such exist. (ii) An upper bound on the dimension of the centroid that is essentially attained by our examples. (iii) The development of a geometric technique to write down border rank decomposition of tensors using centroids and "extended centroids". (iv) The technique is applied to tensors of this paper to prove they are of minimal border rank. The technique is versatile and enables us to geometrically derive and improve upon previous ad hoc decompositions. (v) The construction of symmetric tensors with large centroids and proof that they are wild in the sense of Buczyńska-Buczyński. Our results also pave the way for new upper bounds on the exponent of matrix multiplication. The geometric technique also constructs new "better" tensors for Strassen's laser method from old, and we apply this to the tensors of Strassen and Schönhage to get better tensors in the sense that they give better upper bounds on the exponent than the original tensors.