Fixed Point Strategies in Data Science

TL;DR

This paper advocates for fixed point strategies as a unifying framework in data science, demonstrating their effectiveness in modeling, analyzing, and solving diverse problems across multiple domains.

Contribution

It introduces fixed point methods as a comprehensive approach, reviews key algorithms, and discusses enhancements like stochasticity and non-Euclidean metrics for data science applications.

Findings

Fixed point strategies unify various data science problems.

State-of-the-art algorithms for fixed point construction are reviewed.

Applications span signal processing, machine learning, and inverse problems.

Abstract

The goal of this paper is to promote the use of fixed point strategies in data science by showing that they provide a simplifying and unifying framework to model, analyze, and solve a great variety of problems. They are seen to constitute a natural environment to explain the behavior of advanced convex optimization methods as well as of recent nonlinear methods in data science which are formulated in terms of paradigms that go beyond minimization concepts and involve constructs such as Nash equilibria or monotone inclusions. We review the pertinent tools of fixed point theory and describe the main state-of-the-art algorithms for provably convergent fixed point construction. We also incorporate additional ingredients such as stochasticity, block-implementations, and non-Euclidean metrics, which provide further enhancements. Applications to signal and image processing, machine learning, statistics, neural networks, and inverse problems are discussed.