Information Integration, Organization, and Numerical Harmonic Analysis

This chapter provides an overview of recent developments in methodologies for empirical organization of data. It presents a geometric/analytic mathematical framework for learning, which revolves around building a network or a graph whose nodes are observations. The chapter describes a synthesis of a range of ideas from mathematics and machine learning, which addresses the transition from a local similarity model to a global configuration. This is analogous to Newtonian Calculus, which from a local linear model of variability, calculates a global solution to a differential or partial differential equation. The chapter introduces methodologies that perform signal processing on data matrices, enabling functional regression, prediction, denoising, compression fast numerics, and so on. The aim of the chapter is to discuss how a blend of Geometry and Harmonic Analysis generalizes classical tools of Mathematics, to provide a synthesis of many seemingly unrelated approaches to data analysis and processing.