Linearly-scalable and entropy-optimal learning of nonstationary and nonlinear manifolds

TL;DR

The paper introduces EOMC, a scalable and entropy-optimal manifold clustering method that improves data compression and predictive modeling in complex dynamical systems like fluid mechanics and geosciences.

Contribution

It presents a novel entropy-optimal manifold clustering algorithm with linear complexity, enhancing robustness and efficiency in nonstationary, nonlinear data analysis.

Findings

EOMC effectively captures metastable regime-switching dynamics in chaotic systems.

It achieves significantly lower compression loss compared to PCA-based methods.

Prediction horizons are extended despite chaotic behavior, with slow decrease in mean exit and relaxation times.

Abstract

We propose an Entropy-Optimal Manifold Clustering (EOMC) - and show that it mitigates the cost scaling and robustness issues of the existing dimensionality reduction and manifold learning tools in nonstationary and nonlinear situations, while pertaining the favourable O(T) iteration complexity scaling in the statistics size T, and allowing explicit computation of input data reliability. Application to the Lorenz-96 dynamical system in chaotic regime, as well as to a modified Hasegawa-Wakatani (mHW) model of drift-wave turbulence in the edge of a tokamak plasma reveals that for both of the models their essential dynamics is best described as a metastable regime-switching process, making infrequent transitions between the very persistent low-dimensional manifolds. At the same time, the Markovian mean exit times and relaxation times (that bound the predictability horizons for the identified regime-switching process) appear to decrease only very slowly with the growing external forcing - indicating approximately two-fold longer prediction horizons then is currently anticipated based on analysis of positive Lyapunov exponents, even in very chaotic model regimes. It is also demonstrated that when applied for a lossy compression of the Lorenz-96 and mHW output data in various forcing regimes, EOMC achieves several orders of magnitude smaller compression loss - when compared to the common PCA-related linear compression approaches that build a backbone of the state-of-the-art lossy data compression tools (like JPEG, MP3, and others). These findings open new exciting opportunities for EOMC and transfer operator theory, by offering new possibilities to significantly improve predictive skills and performance of data-driven tools in fluid mechanics and geosciences applications.