Learning in Function Spaces: An Unified Functional Analytic View of Supervised and Unsupervised Learning

TL;DR

This paper presents a unified conceptual framework for supervised and unsupervised learning, interpreting them as variational optimization over data-induced function spaces, clarifying their underlying similarities and differences.

Contribution

It introduces a novel functional analytic perspective that unifies various learning paradigms through data-induced operators and function spaces, enhancing conceptual understanding.

Findings

Connects kernel methods, spectral clustering, and manifold learning under a common framework

Highlights the role of data-induced operators in defining function representations

Clarifies the distinction between learning paradigms based on the choice of functional

Abstract

Many machine learning algorithms can be interpreted as procedures for estimating functions defined on the data distribution. In this paper we present a conceptual framework that formulates a wide range of learning problems as variational optimization over function spaces induced by the data distribution. Within this framework the data distribution defines operators that capture structural properties of the data, such as similarity relations or statistical dependencies. Learning algorithms can then be viewed as estimating functions expressed in bases determined by these operators. This perspective provides a unified way to interpret several learning paradigms. In supervised learning the objective functional is defined using labeled data and typically corresponds to minimizing prediction risk, whereas unsupervised learning relies on structural properties of the input distribution and leads to objectives based on similarity or smoothness constraints. From this viewpoint, the distinction between learning paradigms arises primarily from the choice of the functional being optimized rather than from the underlying function space. We illustrate this framework by discussing connections with kernel methods, spectral clustering, and manifold learning, highlighting how operators induced by data distributions naturally define function representations used by learning algorithms. The goal of this work is not to introduce a new algorithm but to provide a conceptual framework that clarifies the role of function spaces and operators in modern machine learning.