Finite linear least squares is one of the core problems of numerical linear algebra, with countless applications across science and engineering. Consequently, there is a rich and ongoing literature on algorithms for solving linear least squares problems. In this paper, we explore a variant in which the system's matrix has one infinite dimension (i.e., it is a quasimatrix). We call such problems semi-infinite linear regression problems. As we show, the semi-infinite case arises in several applications, such as supervised learning and function approximation, and allows for novel interpretations of existing algorithms. We explore semi-infinite linear regression rigorously and algorithmically. To that end, we give a formal framework for working with quasimatrices, and generalize several algorithms designed for the finite problem to the infinite case. Finally, we suggest the use of various sampling methods for obtaining an approximate solution.
翻译：线性最小方形是数字线性代数的核心问题之一,在科学和工程方面有着无数的应用。 因此,在解决线性最小方形问题的算法方面,有丰富而不断的文献。 在本文中,我们探讨了一个变方,在这个变方中,系统的矩阵有一个无限的维度(即它是一个准矩阵)。我们称之为半无穷线性线性回归问题。正如我们所显示的那样,半无穷性案例出现在几个应用中,例如监督学习和功能近似,并允许对现有算法进行新的解释。我们从逻辑上严格地探索半无穷线性线性回归。为此,我们给出了与准矩阵合作的正式框架,并将为有限问题设计的几种算法推广到无限。最后,我们建议使用各种抽样方法来获得近似的解决办法。
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