Least-Squares and Low-Rank Approximation for Linear Relations Using a Diagrammatic Language
2026-08-24 • Symbolic Computation
Symbolic Computation
AI summaryⓘ
The authors use linear relations, which are a way to think about connections between vectors, to study optimization problems in linear algebra. They show how a relational version of the pseudo-inverse—a tool for finding the best approximate solutions—can be linked to a generalized least-squares problem. Using this, they prove the pseudo-inverse solves certain optimization problems involving these relations. Their main finding is that by trimming this pseudo-inverse in a specific way, they can solve a relational version of the low-rank approximation problem, which includes famous results like the Eckart-Young Theorem and extends to more complex matrix problems.
linear relationspseudo-inverseleast-squares problemoptimizationlow-rank approximationEckart-Young Theoremmatricesvector spaceslinear algebrarelational algebra
Authors
Júlia de Araújo Mota, Iago Leal de Freitas, Lucas Rufino, João Paixão
Abstract
We employ the machinery of linear relations to the study of optimization problems in linear algebra. We first show that the relational version of the pseudo-inverse can be realized through a generalization of the least-squares problem. This allows one to prove that the pseudo-inverse realizes the solution of certain relational optimization problems. Our main result is showing that a certain truncation of this pseudo-inverse defines a solution to a relational version of the classical low-rank approximation problem which recovers both the Eckart-Young Theorem and several optimization problems involving pairs of matrices and vector spaces.