Automated Tensor-Relational Decomposition for Large-Scale Sparse Tensor Computation

TL;DR

This paper introduces a tensor-relational computation framework that enables large-scale sparse tensor operations to be efficiently executed within relational systems, leveraging optimized numerical kernels and automatic rewriting techniques.

Contribution

It presents exttt{EinSum}, a tensor-relational extension of Einstein Summation Notation, and a method to automatically rewrite tensor computations for efficient execution.

Findings

Enables large-scale sparse tensor computations within relational systems.

Automates rewriting of tensor operations into efficient numerical kernels.

Improves handling of sparsity in tensor computations.

Abstract

A \emph{tensor-relational} computation is a relational computation where individual tuples carry vectors, matrices, or higher-dimensional arrays. An advantage of tensor-relational computation is that the overall computation can be executed on top of a relational system, inheriting the system's ability to automatically handle very large inputs with high levels of sparsity while high-performance kernels (such as optimized matrix-matrix multiplication codes) can be used to perform most of the underlying mathematical operations. In this paper, we introduce upper-case-lower-case \texttt{EinSum}, which is a tensor-relational version of the classical Einstein Summation Notation. We study how to automatically rewrite a computation in Einstein Notation into upper-case-lower-case \texttt{EinSum} so that computationally intensive components are executed using efficient numerical kernels, while sparsity is managed relationally.