In this paper we study a worst case to average case reduction for the problem of matrix multiplication over finite fields. Suppose we have an efficient average case algorithm, that given two random matrices $A,B$ outputs a matrix that has a non-trivial correlation with their product $A \cdot B$. Can we transform it into a worst case algorithm, that outputs the correct answer for all inputs without incurring a significant overhead in the running time? We present two results in this direction. (1) Two-sided error in the high agreement regime: We begin with a brief remark about a reduction for high agreement algorithms, i.e., an algorithm which agrees with the correct output on a large (say $>0.9$) fraction of entries, and show that the standard self-correction of linearity allows us to transform such algorithms into algorithms that work in worst case. (2) One-sided error in the low agreement regime: Focusing on average case algorithms with one-sided error, we show that over $\mathbb{F}_2$ there is a reduction that gets an $O(T)$ time average case algorithm that given a random input $A,B$ outputs a matrix that agrees with $A \cdot B$ on at least $51\%$ of the entries (i.e., has only a slight advantage over the trivial algorithm), and transforms it into an $\widetilde{O}(T)$ time worst case algorithm, that outputs the correct answer for all inputs with high probability.

翻译：本文研究了有限域上矩阵乘法问题的从最坏情况到平均情况的降阶。假设存在一个高效的平均情况算法，对于给定的两个随机矩阵 $A,B$，该算法能够输出一个与它们的乘积 $A \cdot B$ 具有非平凡相关性的矩阵。我们能否将其转化为最坏情况算法，使其在不显著增加运行时间的情况下为所有输入输出正确结果？我们在此方向上提出两项结果：(1) 高一致性区间内的双边误差：我们首先简要讨论了高一致性算法的降阶问题，即算法在大部分（例如大于0.9）矩阵元素上与正确输出一致，并证明标准线性自校正方法可以将此类算法转化为适用于最坏情况的算法。(2) 低一致性区间内的单边误差：针对单边误差的平均情况算法，我们证明在域 $\mathbb{F}_2$ 上存在一种降阶方法，能够将运行时间为 $O(T)$ 的平均情况算法（该算法在给定随机输入 $A,B$ 时，输出矩阵与 $A \cdot B$ 在至少51%的元素上一致，即仅比平凡算法略优）转化为运行时间为 $\widetilde{O}(T)$ 的最坏情况算法，且该算法能以高概率为所有输入输出正确结果。