Topology-Based Machine Learning and Regime Identification in Stochastic, Heavy-Tailed Financial Time Series

Classic machine learning and regime identification methods applied to financial time series lack theoretical guarantees and exhibit systematic failure modes: heavy-tails invalidate moment-based geometry, rendering distances and centroids dominated by extremes or unstable; jumps violate smoothness, destabilizing local regressions, kernel methods, and gradient-based learning; and non-stationarity disrupts neighborhood relations, so distances in classical feature spaces no longer reflect meaningful proximity. To address these challenges, we propose a topology-based machine-learning framework grounded on probabilistic reconstruction of state-space geometry, which replaces moment- and smoothness-dependent representations with deformation-stable summaries of state-space geometry, preserving neighborhoods, adjacency, and topology. The finite-sample validity of homeomorphic state-space reconstruction, required for topology-based machine learning, is assessed through numerical studies on synthetic data with heavy tails, jumps, and known ground-truth regimes. Further diagnostics of local invertibility and bounded geometric distortion quantify when embedding windows are consistent with local diffeomorphic behavior, enabling metric-sensitive, geometry-aware learning. Clustering of Hilbert-space summaries accurately recovers underlying market tail-risk regimes with robust results across selected filtrations. Temporal, feature-space, and cluster-label null tests confirm that topology-based clustering captures genuine topological structure rather than noise or artifacts, and encodes temporal dependencies at local, mesoscopic, and network levels associated with market regimes.