Relevant, hidden, and frustrated information in high-dimensional analyses of complex dynamical systems with internal noise

TL;DR

The paper questions the assumption that high-dimensional analyses are always necessary for understanding complex dynamical systems with internal noise. It combines SOAP-based high-dimensional descriptors with time-series Onion Clustering to extract information from water/ice coexistence trajectories, validating a second dataset from Quincke roller experiments to test generality. A key finding is that a single SOAP component accounting for $<0.001%$ of the variance can distinguish ice, water, and the ice–water interface, while adding more dimensions often introduces noise and leads to information loss, a phenomenon termed frustrated information; oversampling can even cause information hallucinations. The results challenge the assumption that higher dimensionality improves understanding and advocate focusing on information quality and the identification of physically relevant dimensions and optimal time-resolutions for robust analyses across systems.

Abstract

Extracting from trajectory data meaningful information to understand complex molecular systems might be non-trivial. 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 the Smooth Overlap of Atomic Positions (SOAP) descriptor, obtaining a large dataset containing 2.56e6 576-dimensional SOAP vectors 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.

Figures (13)

• Figure 1: Ice/water coexistence MD simulation. a) Screenshot of the system studied. b) Schematic representation of the steps necessary to extract information from simulated systems. c) Zoom into water molecules, centered in a molecule (red) within a cutoff $r$. d) Zoom of the water molecules, considering exclusively oxygen atoms, which are used to calculate the SOAP descriptor. e) Definition of the Smooth Overlap of Atomic Position (SOAP) descriptor with distinction of components with $l = 0$ and $l > 0$.
• Figure 2: The information nested along the time dimension: Onion Clustering on PC1 and PC2 time-series data. a) Top: Cumulative data variance explained by the first four PCs. Bottom: SOAP dataset projection onto the first 2 PCs. Red contour isolines help to visualize the data density. Inset: common clustering approaches (hierarchical clustering) distinguish two main clusters (solid ice and liquid water) in the SOAP dataset. b) Left: PC1 time-series data of all the molecules. Center: kernel density estimate (KDE) of the PC1 time-series data, with the Gaussian environments (solid curves) identified by Onion Clustering Becchi2024layer. The dark gray environment corresponds to solid ice, the red one to the ice/water interface, and the blue one to liquid water. Right: snapshot of the MD simulation, where the water molecules are colored according to the Onion micro-clusters. c) Onion output plot, indicating the number of clusters (in blue) identified by Onion Clustering, and the fraction of lost information (in orange: unclassified information (ENV0) -- fast SOAP changes -- due to insufficient resolution) as a function of the time-resolution ($\Delta t$) used in the analysis of the PC1 time-series. d) Same as panel c), but for the PC2 time-series. e) Same as panel b), but for the PC2 time-series. f) Left: same results as panel c), obtained by Onion Clustering analyzing a bivariate (bi-dimensional) PC1, PC2 time-series (classified clusters in blue, unclassifiable information in orange). The gray dotted line shows the best result obtained by either of the two single components: the gray area shows the information lost by combining both dimensions (gray = max(#$_{clust}$(PC1),#$_{clust}$(PC2))-#$_{clust}$(PC1,PC2)). Right: snapshot of the MD simulation, where the water molecules are colored according to the Onion micro-clusters detected at the resolution of $\Delta t = 3$ ns (red dotted vertical line) from the bi-dimensional (PC1,PC2) time-series.
• Figure 3: Onion Clustering on denoised PC1 and PC2. The figure follows the same structure of Figure \ref{['fig2']}, but all the analysis are performed on the smoother time-series PC1$_{\text{dn}}$ and PC2$_{\text{dn}}$. The results of PC3$_{\text{dn}}$ and PC4$_{\text{dn}}$ are reported in Supporting Figure SI4
• Figure 4: Onion Clustering on denoised components. a) Variance of the six most significant spherical ($l=0$) SOAP components. b) Variance of the six most significant non-spherical ($l>0$) SOAP components. c) Denoised dataset projection onto components #63 and #237; red contour lines help visualize the data density; in the inset, static clustering distinguishes 2 environments. d), e) Onion Clustering results of component $\#63_{\text{dn}}$. f), g) Onion Clustering results of component $\#237_{\text{dn}}$.
• Figure 5: Information frustration: a) Onion Clustering on bivariate $\#63_{\text{dn}}$ and $\#237_{\text{dn}}$ time-series (see Figure \ref{['fig2']}f for the legend). In gray, the combination of two components leads to considerable information loss, even when they are denoised. b-c) Dependence of the detection of the ice/water interface on the sampling $\Delta t$ in the raw MD trajectories onto which the PC1 is calculated in the native/noisy (b) or denoised (c) SOAP dataset.