Generalizations of Matrix Multiplication can solve the Light Bulb Problem

In the light bulb problem, one is given uniformly random vectors $x_1, \ldots, x_n, y_1, \ldots, y_n \in \{-1,1\}^d$. They are all chosen independently except a planted pair $(x_{i^*}, y_{j^*})$ is chosen with correlation $\rho>0$. The goal is to find the planted pair. This problem was introduced over 30 years ago by L.~Valiant, and is known to have many applications in data analysis, statistics, and learning theory. The naive algorithm runs in $\Omega(n^2)$ time, and algorithms based on Locality-Sensitive Hashing approach quadratic time as $\rho \to 0$. In 2012, G.~Valiant gave a breakthrough algorithm using fast matrix multiplication that runs in time $O(n^{(5-\omega)/(4-\omega)}) < O(n^{1.615})$, no matter how small $\rho>0$ is. This was subsequently refined by Karppa, Kaski, and Kohonen in 2016 to $O(n^{2 \omega / 3}) < O(n^{1.582})$. In this paper, we propose a new approach which can replace matrix multiplication tensor with other tensors. Those tensors can omit some terms one is supposed to compute, and include additional error terms. Our new approach can make use of any tensors which previously had no known algorithmic applications, including tensors which arise naturally as intermediate steps in border rank methods and in the Laser method. We further show that our approach can be combined with locality-sensitive hashing to design an algorithm whose running time improves as $\rho$ gets larger. To our knowledge, this is the first algorithm which combines fast matrix multiplication with hashing for the light bulb problem or any closest pair problem, and it leads to faster algorithms for small $\rho>0$. We also introduce a new tensor $T_{2112}$, which has the same size of $2 \times 2$ matrix multiplication tensor, but runs faster than the Strassen's algorithm for light bulb problem.