Measurability and continuity of parametric low-rank approximation in Hilbert spaces: linear operators and random variables

Abstract

We present a unified theoretical framework for parametric low-rank approximation, a research area devoted to the development of efficient algorithms that act as adaptive alternatives of traditional methods such as Singular Value Decomposition (SVD), Proper Orthogonal Decomposition (POD), and Principal Component Analysis (PCA). Applications include, e.g., the numerical treatment of parameter-dependent partial differential equations, where operators vary with parameters, and the statistical analysis of longitudinal data, where complex measurements, like audio signals and images, are collected over time. Recently, several adaptive algorithms have emerged, but a common mathematical foundation is still lacking, and existing solutions remain constrained to specific applications. As a result, key theoretical questions -- such as the existence and regularity of optimal parametric low-rank approximants -- remain inadequately addressed. Our goal is to bridge this gap between theory and practice by establishing a rigorous framework for parametric low-rank approximation under minimal assumptions, specifically focusing on cases where parameterizations are either measurable or continuous. The analysis is carried out within the context of separable Hilbert spaces, ensuring applicability to both finite and infinite-dimensional settings. Finally, connections to recently emerging trends in the Deep Learning literature, relevant for engineering and data science, are also discussed.