Relevant, Hidden, and Frustrated Information in High-Dimensional Analyses of Complex Dynamical Systems with Internal Noise

Extracting from trajectory data meaningful information to understand complex molecular systems might be nontrivial. High-dimensional analyses are typically assumed to be desirable, if not required, to prevent losing important information. But to what extent such high-dimensionality is really needed/beneficial often remains unclear. Here we challenge such a fundamental general problem. As a representative case of a system with internal dynamical complexity, we study atomistic molecular dynamics trajectories of liquid water and ice coexisting in dynamical equilibrium at the solid/liquid transition temperature. To attain an intrinsically high-dimensional analysis, we use as an example an abstract high-dimensional descriptor of local molecular environments (e.g., Smooth Overlap of Atomic Positions, SOAP), obtaining a large dataset containing 2.56 × 106 576-dimensional SOAP spectra that we analyze in various ways. Our results demonstrate how the time-series data contained in one single SOAP dimension accounting only <0.001% of the total dataset's variance (neglected and discarded in typical variance-based dimensionality reduction approaches) allows resolving a remarkable amount of information, classifying/discriminating the bulk of water and ice phases, as well as two solid-interface and liquid-interface layers as four statistically distinct dynamical molecular environments. Adding more dimensions to this one is found not only ineffective but even detrimental to the analysis due to recurrent negligible-information/non-negligible-noise additions and "frustrated information" phenomena leading to information loss. Such effects are proven general and are observed also in completely different systems and descriptors' combinations. This shows how high-dimensional analyses are not necessarily better than low-dimensional ones to elucidate the internal complexity of physical/chemical systems, especially when these are characterized by non-negligible internal noise.