Date: November 22, 2018 16:15–17:00

Place: Exactum C122 (Kumpula)

Speaker: Petteri Kaski

**Title**: Probabilistic tensors and opportunistic Boolean matrix multiplication

**Abstract**: We study probabilistic extensions of classical deterministic measures of algebraic complexity of a tensor, such as the rank and the border rank. These probabilistic extensions enable improvements over their deterministic counterparts for specific tensors of interest, starting from the tensor $\langle 2,2,2 \rangle$ that represents $2\times 2$ matrix multiplication. While it is well known that the (deterministic) tensor rank and border rank of $\langle 2,2,2 \rangle$ is 7 [V. Strassen, Numer. Math. 13 (1969); J. E. Hopcroft and L. R. Kerr, SIAM J. Appl. Math. 20 (1971); S. Winograd, Linear Algebra Appl. 4 (1971); J. M. Landsberg, J. AMS 19 (2006)] we show that the probabilistic rank of $\langle 2,2,2 \rangle$ is at most 6+\frac{6}{7} and the probabilistic border rank is at most 6+\frac{2}{3}. By submultiplicativity, this leads immediately to novel randomized algorithm designs, such as algorithms for Boolean matrix multiplication as well as detecting and estimating the number of triangles in graphs.

Joint work with Matti Karppa (Aalto University).