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

2026-03-09

AI Summary

This work addresses the inefficiency of large-scale sparse tensor computations by proposing a tensor-relational hybrid computation model. The model introduces a novel case-sensitive Einstein summation (EinSum) notation that automatically decomposes sparse tensor expressions into a combination of relational algebra operations and dense tensor computations: sparsity is efficiently managed by a relational database system, while dense numerical kernels handle the compute-intensive parts. This approach enables co-optimization of sparse structure handling and high-performance computing, significantly improving overall performance without sacrificing sparse processing efficiency. It represents the first end-to-end automatic compiler capable of translating high-level EinSum expressions into efficient hybrid execution plans.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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.

Problem

• tensor-relational computation
• Einstein Summation Notation
• sparse tensor
• automated decomposition
• large-scale computation

Innovation

• tensor-relational decomposition
• Einstein Summation
• sparse tensor computation
• automatic rewriting
• relational systems