Spectral invariants of ergodic symbolic systems for pattern recognition and anomaly detection.

Abstract

Despite tangible advances in machine learning (ML) over the last few decades, many of the ML techniques still suffer from fundamental issues like overfitting and lack of explainability. These issues mandate requirements for mathematical rigor to ensure robust learning from observed data. In this context, topological invariants in data manifolds provide a rich representation of the underlying dynamical system, which can be utilized for developing a mathematically rigorous ML tool to characterize the dynamical behaviour and operational phases of the underlying process. This paper aims to investigate spectral invariants of symbolic systems for detecting changes in topological characteristics of data manifolds. A novel ML approach is proposed, where commutator norms are used on sequences of endomorphisms to symbolically describe dynamical systems on probability spaces with ergodic measures. The objective here is to detect topological invariants of data manifolds that can be used for signal processing, pattern recognition, and anomaly detection. The proposed ML approach is validated on models of selected chaotic dynamical systems for prompt detection of phase transitions. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.