A Hilbert Space of Stationary Ergodic Processes

TL;DR

This paper develops a rigorous mathematical framework that models stationary ergodic processes as a Hilbert space, enabling noise-resistant analysis and potential applications in classification and clustering.

Contribution

It introduces a complete inner product structure on the space of ergodic stationary processes, facilitating geometric analysis of data streams under noise.

Findings

Angles between data streams are preserved under noise

Orthogonality of stochastic processes is rigorously defined

Framework supports noise-robust data analysis techniques

Abstract

Identifying meaningful signal buried in noise is a problem of interest arising in diverse scenarios of data-driven modeling. We present here a theoretical framework for exploiting intrinsic geometry in data that resists noise corruption, and might be identifiable under severe obfuscation. Our approach is based on uncovering a valid complete inner product on the space of ergodic stationary finite valued processes, providing the latter with the structure of a Hilbert space on the real field. This rigorous construction, based on non-standard generalizations of the notions of sum and scalar multiplication of finite dimensional probability vectors, allows us to meaningfully talk about "angles" between data streams and data sources, and, make precise the notion of orthogonal stochastic processes. In particular, the relative angles appear to be preserved, and identifiable, under severe noise, and will be developed in future as the underlying principle for robust classification, clustering and unsupervised featurization algorithms.