发表机构
University of Zurich（苏黎世大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对数据库连接上线性代数运算的数值稳定性问题，本文提出投影后向稳定性与数据库条件数，量化扰动放大并建立与经典后向稳定性的联系。
AI 中文摘要
数据库文献中的大量工作为关系连接定义的矩阵上的线性代数和机器学习开发了高效算法，然而此类计算的数值稳定性至今未受到关注。这是一个实际问题：连接矩阵可能远大于输入数据库，且其中包含的输入值的重复副本会放大对其执行的数值运算所产生的浮点误差。本文首次对数据库连接上的线性代数数值稳定性进行形式化研究。我们首先证明，数值稳定性的标准衡量标准——后向稳定性——在此情境下失效：连接矩阵构成环境矩阵空间的一个结构化子空间，因此解释计算结果所需的扰动未必对应于任何扰动的输入数据库。这种失效甚至出现在像矩阵-向量乘法这样简单的运算中。为克服此局限，我们引入投影后向稳定性，它是将后向稳定性从单函数计算推广到两个函数复合计算的一种泛化，并建立了其与经典后向稳定性的联系。在我们的数据库情境中，这两个函数分别是连接查询和数值运算。我们进一步引入数据库条件数，定义为输入数据值复制到连接矩阵中的最大副本数与最小副本数之比的平方根，并证明它量化了连接矩阵的扰动如何被放大为输入数据库的扰动，且与所用计算无关。数据库条件数与将输入值复制到连接矩阵中的扩展矩阵的经典条件数一致。
英文摘要
A large body of work in the database literature develops efficient algorithms for linear algebra and machine learning over matrices defined by relational joins, yet the numerical stability of such computations has so far received no attention. This is a practical concern: a join matrix can be much larger than the input database, and the repeated copies of input values it contains compound the floating-point errors incurred by the numerical operations performed over it. This paper initiates a formal investigation of numerical stability for linear algebra over database joins. We first show that backward stability, the standard yardstick of numerical stability, loses its effectiveness in this setting: join matrices form a structured subspace of the ambient matrix space, so a perturbation explaining a computed result need not correspond to any perturbed input database. This failure already occurs for operations as simple as matrix-vector multiplication. To overcome this limitation, we introduce projected backward stability, a generalization of backward stability from the computation of one function to that of a composition of two functions, and establish its connection to classical backward stability. In our database setting, the two functions are the join query and the numerical operation. We further introduce the database condition number as the square root of the ratio of maximal to minimal number of copies of input data values into the join matrix, and show that it quantifies how a perturbation of the join matrix is amplified into a perturbation of the input database, independently of the computation used. The database condition number coincides with the classical condition number of the expansion matrix that replicates input values into the join matrix.
