A Refined Laser Method and Faster Matrix Multiplication (2010.05846v2)

Abstract: The complexity of matrix multiplication is measured in terms of $\omega$, the smallest real number such that two $n\times n$ matrices can be multiplied using $O(n^{\omega+\epsilon})$ field operations for all $\epsilon>0$; the best bound until now is $\omega<2.37287$ [Le Gall'14]. All bounds on $\omega$ since 1986 have been obtained using the so-called laser method, a way to lower-bound the `value' of a tensor in designing matrix multiplication algorithms. The main result of this paper is a refinement of the laser method that improves the resulting value bound for most sufficiently large tensors. Thus, even before computing any specific values, it is clear that we achieve an improved bound on $\omega$, and we indeed obtain the best bound on $\omega$ to date: $$\omega < 2.37286.$$ The improvement is of the same magnitude as the improvement that [Le Gall'14] obtained over the previous bound [Vassilevska W.'12]. Our improvement to the laser method is quite general, and we believe it will have further applications in arithmetic complexity.

• The paper refines the laser method to improve the upper bound of ω from 2.37287 to 2.37286.
• It enhances tensor evaluation by applying nuanced partitioning and Salem-Spencer set constructions on Coppersmith–Winograd tensors.
• The refined method paves the way for more efficient matrix multiplication algorithms and advances computational complexity research.

A Refined Laser Method and Faster Matrix Multiplication

The paper presents advancements in the theoretical understanding of matrix multiplication complexity through a refined laser method, which offers an improved bound on the exponent of matrix multiplication, $\omega$. This work builds upon decades of progress in decreasing the value of $\omega$, defined as the smallest real number satisfying that two $n \times n$ matrices can be multiplied using $O(n^{\omega+\epsilon})$ operations for any $\epsilon > 0$. The most significant contribution of this paper is improving the upper bound of $\omega$ from 2.37287 to 2.37286.

Overview of the Refined Laser Method

The laser method is instrumental in analyzing matrix multiplication tensors. It functions by lower bounding the “value” of a tensor, assisting in the design of matrix multiplication algorithms. This paper introduces a refined version of the laser method that enhances the value bound for sufficiently large tensors, thus impacting the bound on $\omega$.

The keystone improvement is demonstrated in the evaluation of a tensor's value. The new technique allows for a better interpretation of the tensor partitions, leading to stronger value bounds by an order inherent in the tensor structure. This advancement directly leads to a reduction in $\omega$, moving it to an unprecedented lower bound of 2.37286.

Strong Numerical Results

The paper utilizes the refined laser method to specifically analyze powers of the Coppersmith-Winograd tensor, $CW_q$. Through careful examination and selection of these tensors, the authors demonstrate a bound on $\omega$ that benefits from these structural refinements. The process includes leveraging Salem-Spencer set constructions, which facilitates the zeroing-out method, further enhancing the tensor value understanding.

A significant part of the numerical success of the refined method lies in its application to $CW_5^{\otimes 32}$, where the final bound obtained, $\omega<2.3728596$, is computed by achieving a high value bound on the tensor.

Implications and Future Directions

The implications of reducing the upper limit of $\omega$ are both practical and theoretical. Practically, this opens avenues for designing more efficient matrix multiplication algorithms, which have broad applications in scientific computing and data processing tasks. Theoretically, refining the laser method adds to the computational complexity field, advancing our understanding of arithmetic complexity.

Future research can explore further applications of this refined laser method across different tensors in computational mathematics and beyond. The potential for more significant reductions or different applications in tensor computations could drive further pursuit. Additionally, examining whether new tensor families could exploit the refined method to surpass current limitations presents an open research trajectory.

Conclusion

This paper presents a meticulous refinement of the laser method applied to matrix multiplication, setting a new benchmark for the upper bound of $\omega$. This advancement exemplifies how nuanced improvements in theoretical techniques can yield measurable impacts in computational complexity. The extension of these results could hold promise for broader future applications and ongoing innovation within the field.