Communication Lower Bounds and Optimal Algorithms for Multiple Tensor-Times-Matrix Computation

TL;DR

This paper establishes fundamental communication lower bounds for Multi-TTM tensor computations and proposes an optimal parallel algorithm that minimizes data movement, improving efficiency in multidimensional data analysis.

Contribution

It derives the first communication lower bounds for Multi-TTM and introduces a parallel algorithm that achieves these bounds, optimizing data movement in tensor computations.

Findings

Communication lower bounds are established for Multi-TTM.

The proposed algorithm attains these bounds, ensuring optimal data movement.

The new algorithm reduces communication compared to naive approaches.

Abstract

Multiple Tensor-Times-Matrix (Multi-TTM) is a key computation in algorithms for computing and operating with the Tucker tensor decomposition, which is frequently used in multidimensional data analysis. We establish communication lower bounds that determine how much data movement is required to perform the Multi-TTM computation in parallel. The crux of the proof relies on analytically solving a constrained, nonlinear optimization problem. We also present a parallel algorithm to perform this computation that organizes the processors into a logical grid with twice as many modes as the input tensor. We show that with correct choices of grid dimensions, the communication cost of the algorithm attains the lower bounds and is therefore communication optimal. Finally, we show that our algorithm can significantly reduce communication compared to the straightforward approach of expressing the computation as a sequence of tensor-times-matrix operations.