Fast Matrix Multiplication Without Tears: A Constraint Programming Approach

It is known that the multiplication of an $N \times M$ matrix with an $M \times P$ matrix can be performed using fewer multiplications than what the naive $NMP$ approach suggests. The most famous instance of this is Strassen's algorithm for multiplying two $2\times 2$ matrices in 7 instead of 8 multiplications. This gives rise to the constraint satisfaction problem of fast matrix multiplication, where a set of $R < NMP$ multiplication terms must be chosen and combined such that they satisfy correctness constraints on the output matrix. Despite its highly combinatorial nature, this problem has not been exhaustively examined from that perspective, as evidenced for example by the recent deep reinforcement learning approach of AlphaTensor. In this work, we propose a simple yet novel Constraint Programming approach to find non-commutative algorithms for fast matrix multiplication or provide proof of infeasibility otherwise. We propose a set of symmetry-breaking constraints and valid inequalities that are particularly helpful in proving infeasibility. On the feasible side, we find that exploiting solver performance variability in conjunction with a sparsity-based problem decomposition enables finding solutions for larger (feasible) instances of fast matrix multiplication. Our experimental results using CP Optimizer demonstrate that we can find fast matrix multiplication algorithms for matrices up to $3\times 3$ in a short amount of time

An adaptive prefix-assignment technique for symmetry reduction

This paper presents a technique for symmetry reduction that adaptively assigns a prefix of variables in a system of constraints so that the generated prefix-assignments are pairwise nonisomorphic under the action of the symmetry group of the system. The technique is based on McKay's canonical extension framework [J.~Algorithms 26 (1998), no.~2, 306--324]. Among key features of the technique are (i) adaptability---the prefix sequence can be user-prescribed and truncated for compatibility with the group of symmetries; (ii) parallelizability---prefix-assignments can be processed in parallel independently of each other; (iii) versatility---the method is applicable whenever the group of symmetries can be concisely represented as the automorphism group of a vertex-colored graph; and (iv) implementability---the method can be implemented relying on a canonical labeling map for vertex-colored graphs as the only nontrivial subroutine. To demonstrate the practical applicability of our technique, we have prepared an experimental open-source implementation of the technique and carry out a set of experiments that demonstrate ability to reduce symmetry on hard instances. Furthermore, we demonstrate that the implementation effectively parallelizes to compute clusters with multiple nodes via a message-passing interface.Comment: Updated manuscript submitted for revie

Improving the numerical stability of fast matrix multiplication

Fast algorithms for matrix multiplication, namely those that perform asymptotically fewer scalar operations than the classical algorithm, have been considered primarily of theoretical interest. Apart from Strassen's original algorithm, few fast algorithms have been efficiently implemented or used in practical applications. However, there exist many practical alternatives to Strassen's algorithm with varying performance and numerical properties. Fast algorithms are known to be numerically stable, but because their error bounds are slightly weaker than the classical algorithm, they are not used even in cases where they provide a performance benefit. We argue in this paper that the numerical sacrifice of fast algorithms, particularly for the typical use cases of practical algorithms, is not prohibitive, and we explore ways to improve the accuracy both theoretically and empirically. The numerical accuracy of fast matrix multiplication depends on properties of the algorithm and of the input matrices, and we consider both contributions independently. We generalize and tighten previous error analyses of fast algorithms and compare their properties. We discuss algorithmic techniques for improving the error guarantees from two perspectives: manipulating the algorithms, and reducing input anomalies by various forms of diagonal scaling. Finally, we benchmark performance and demonstrate our improved numerical accuracy