使用图示语言处理线性关系的最小二乘与低秩近似
Least-Squares and Low-Rank Approximation for Linear Relations Using a Diagrammatic Language
AI总结：
该研究将线性关系机制应用于线性代数优化，推广最小二乘实现伪逆的关系版本，证明其可解关系型优化，还得到伪逆截断形式可解含Eckart-Young定理等的低秩近似关系版本。
AI中文摘要：
我们将线性关系的机制应用于线性代数中的优化问题研究。首先，我们证明伪逆的关系版本可通过最小二乘问题的推广实现，这使得可以证明伪逆能求解某些关系型优化问题。我们的主要结果是，该伪逆的某种截断形式可定义经典低秩近似问题的关系版本的解，该解涵盖了Eckart-Young定理以及涉及矩阵对和向量空间的若干优化问题。
英文摘要：
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.
