On the Expressiveness of LARA: A Unified Language for Linear and Relational Algebra

TL;DR

This paper analyzes the expressive power of LARA, a unified language for relational and linear algebra, showing its capabilities and limitations in expressing various matrix operations relevant to database and machine learning tasks.

Contribution

It characterizes LARA's expressive completeness, explores how different assumptions affect its ability to express key matrix operations, and discusses the implications for database and machine learning applications.

Findings

LARA is expressively complete with respect to first-order logic with aggregation.

Under strong genericity, LARA cannot express matrix convolution or inverse.

Adding an order allows convolution but complicates understanding the language's expressive power.

Abstract

We study the expressive power of the LARA language -- a recently proposed unified model for expressing relational and linear algebra operations -- both in terms of traditional database query languages and some analytic tasks often performed in machine learning pipelines. We start by showing LARA to be expressive complete with respect to first-order logic with aggregation. Since LARA is parameterized by a set of user-defined functions which allow to transform values in tables, the exact expressive power of the language depends on how these functions are defined. We distinguish two main cases depending on the level of genericity queries are enforced to satisfy. Under strong genericity assumptions the language cannot express matrix convolution, a very important operation in current machine learning operations. This language is also local, and thus cannot express operations such as matrix inverse that exhibit a recursive behavior. For expressing convolution, one can relax the genericity requirement by adding an underlying linear order on the domain. This, however, destroys locality and turns the expressive power of the language much more difficult to understand. In particular, although under complexity assumptions the resulting language can still not express matrix inverse, a proof of this fact without such assumptions seems challenging to obtain.