A more accurate rational non-commutative algorithm for multiplying 4x4 matrices using 48 multiplications

Abstract: We propose a more accurate variant of an algorithm for multiplying 4x4 matrices using 48 multiplications over any ring containing an inverse of 2. This algorithm has an error bound exponent of only log 4 $γ$\infty$,2$\approx$ 2.386. It also reaches a better accuracy w.r.t. max-norm in practice, when compared to previously known such fast algorithms. Furthermore, we propose a straight line program of this algorithm, giving a leading constant in its complexity bound of 387 32 n 2+log 4 3 + o n 2+log 4 3 operations over any ring containing an inverse of 2. Introduction: An algorithm to multiply two 4x4 complex-valued matrices requiring only 48 non-commutative multiplications was introduced in [16] 1 using a pipeline of LLMs orchestrated by an evolutionary coding agent. A matrix multiplication algorithm with that many non-commutative multiplications is denoted by 4x4x4:48 in the sequel. An equivalent variant of the associated tensor decomposition defining this algorithm, but over the rationals (more precisely over any ring containing an inverse of 2), was then given in [8]. Most error analysis of sub-cubic time matrix multiplication algorithms [3, 4, 2, 1, 17] are given in the max-norm setting: bounding the largest output error as a function of the max-norm product of the vectors of input matrix coefficients. In this setting, Strassen's algorithm has shown the best accuracy bound, (proven minimal under some assumptions in [2]). In [6, 8], the authors relaxed this setting by shifting the focus to the 2-norm for input and/or output; that allowed them to propose a 2x2x2:7 variant with an improved accuracy bound. Experiments show that this variant performs best even when measuring the max-norm of the error bound. We present in this note a variant of the recent 4x4x4:48 algorithm over the rationals (again in the same orbit under De Groot isotropies [10]) that is more numerically accurate w.r.t. max-norm in practice. In particular, our new variant improves on the error bound exponent, from log 2 $γ$$\infty$,2$\approx$ 2.577 Consider the product of an M x K matrix A by a K x N matrix B. It is computed by a m, k, n algorithm represented by the matrices L, R, P applied recursively on ${\ell}$$recursive levels and the resulting m 0 x k 0 by k 0 x n 0 products are performed using an algorithm$$β$. Here M = m 0 m${\ell}$, K = k 0 k${\ell}$and n = n 0 n${\ell}$$. The accuracy bound below uses any (possibly different) p-norms and q-norms for its left-handside, <strong><em>$$\bullet$</em></strong> p and right-hand side, <strong><em>$\bullet$$</em></strong> q . The associated dual norms, are denoted by <strong><em>$$\bullet$</em></strong> p$\star$and <strong><em>$\bullet$</em></strong> q$\star$ respectively. Note that, these are vector norms, hence A p for matrix A in R mxn denotes Vect(A) p and is the p-norm of the mn dimensional vector of its coefficients, and not a matrix norm.