Neural Persistence Dynamics

TL;DR

Various (ablation) experiments not only demonstrate the relevance of each model component but provide compelling empirical evidence that the proposed model - Neural Persistence Dynamics - substantially outperforms the state-of-the-art across a diverse set of parameter regression tasks.

Abstract

We consider the problem of learning the dynamics in the topology of time-evolving point clouds, the prevalent spatiotemporal model for systems exhibiting collective behavior, such as swarms of insects and birds or particles in physics. In such systems, patterns emerge from (local) interactions among self-propelled entities. While several well-understood governing equations for motion and interaction exist, they are notoriously difficult to fit to data, as most prior work requires knowledge about individual motion trajectories, i.e., a requirement that is challenging to satisfy with an increasing number of entities. To evade such confounding factors, we investigate collective behavior from a $\textit{topological perspective}$, but instead of summarizing entire observation sequences (as done previously), we propose learning a latent dynamical model from topological features $\textit{per time point}$. The latter is then used to formulate a downstream regression task to predict the parametrization of some a priori specified governing equation. We implement this idea based on a latent ODE learned from vectorized (static) persistence diagrams and show that a combination of recent stability results for persistent homology justifies this modeling choice. Various (ablation) experiments not only demonstrate the relevance of each model component but provide compelling empirical evidence that our proposed model - $\textit{Neural Persistence Dynamics}$ - substantially outperforms the state-of-the-art across a diverse set of parameter regression tasks.

Figures (5)

• Figure 1: Conceptual overview of Neural Persistence Dynamics. Given is a sequence of observed point clouds $\mathcal{P}_{\tau_0},\ldots,\mathcal{P}_{\tau_N}$. First, we summarize each $\mathcal{P}_{\tau_i}$ via (Vietoris-Rips) persistent homology into persistence diagrams (zero-, one- and two-dimensional; only the zero-dimensional diagrams are shown) which are then vectorized into $\mathbf{v}_{\tau_i}$ via existing techniques. Second, we model the dynamics in the sequence $\mathbf{v}_{\tau_0},\ldots,\mathbf{v}_{\tau_N}$ via a continuous latent variable model (in our case, a latent ODE) and then use a summary of the latent path to predict the parameters of specific governing equation(s) of collective behavior. Precomputed steps are highlighted in red.
• Figure 2: Schematic illustration of different model variants. The first three variants (left to right) explicitly model latent dynamics (later denoted as w/ dynamics), the baseline variants do not (later denoted as w/o dynamics), but still incorporate the attention mechanism of the encoder from Shukla21a, which we use throughout.
• Figure 3: Models of collective behavior. Parameters that are varied to obtain different behavior are highlighted in red; the range of each parameter is listed in \ref{['appendix:subsection:simulationsettings']}. In the Vicsek model, $\mathbf{B}_t$ denotes Brownian motion.
• Figure 4: Impact of the maximal simulation time $T$ for extracting training/testing sequences starting at $\tau_{0} \in [0,T-1000]$, assessed on the dorsogna-10k dataset.
• Figure 5: Birth rates $\lambda_b$ and death rates $\lambda_d$ used for generating the volex-10k dataset.

Theorems & Definitions (4)

• Remark 1
• Remark 2
• Remark 3
• Remark 4