Discrimination of Dynamic Data via Curvature Sets

TL;DR

A new algorithm is presented to efficiently compute the erosion distance between arbitrary antichain-decomposable modules between arbitrary antichain-decomposable modules (including, but not limited to modules produced by the construction).

Abstract

Techniques from topological data analysis (TDA) have proven effective in studying time-dependent data arising in dynamic systems, such as animal swarming behavior and spatiotemporal patterns in neuroscience. While early algorithms leveraged efficient updates to persistence diagrams for dynamic data, they struggled to distinguish behaviors that are isometric at each fixed time but differ qualitatively. This limitation was addressed by Kim and Mémoli, who introduced a spatiotemporal persistence framework for dynamic metric spaces, resulting in multiparameter persistence modules. However, these modules pose computational challenges. To address this, we build on insights from Gómez and Mémoli, who observed that the homology of Rips complexes over size $(2k+2)$ point subsets of a metric space--termed principal curvature sets--is both tractable and informative. We extend this idea to dynamic settings by introducing dynamic curvature-set persistent homology, applying the spatiotemporal framework of Kim and Mémoli to curvature sets. We prove that the resulting multiparameter persistence modules are interval-decomposable: in fact, they possess a stronger property we term antichain-decomposable. Utilizing this property, we present a new algorithm to efficiently compute the erosion distance $d_E$ (due to Patel) between arbitrary antichain-decomposable modules (including, but not limited to modules produced by our construction). Additionally, our construction is stable with respect to a generalized Gromov-Hausdorff distance between time-dependent datasets proposed by Kim and Mémoli. This enables a robust computational pipeline for distinguishing dynamic data, as demonstrated in experiments with the Boids model, where we successfully detect parameter changes.

Figures (4)

• Figure 1: Two dynamic metric spaces exhibiting qualitatively different behavior. Here $r>0$, $x_1,x_3,y_1$ and $y_3$ are fixed at the displayed coordinate values whereas $x_2(t) = r\sin t$ and $y_2(t) = r |\sin t|$. For any fixed $t$, the resulting configurations are isometric as metric spaces (when $x_2$ is to the left of the midpoint of $x_1$ and $x_3$, reflect the left space through the midpoint to show isometry). However, the two dynamic behaviors are certainly different. Figure courtesy of Kim and Mémoli; see spatiotemporal.
• Figure 2: Examples of $4$-point metric spaces.
• Figure 3: Visualization of distance matrix between flocks produced by $d_E$-induced distance
• Figure 4: Some cross-polytopes

Theorems & Definitions (80)

• Definition 2.1: Dynamic metric spaces (DMS) stablesignatures
• Definition 2.2: Semimetric space
• Definition 2.3: spatiotemporal
• Definition 2.4: Rips complex
• Definition 2.5: Rips filtration
• Definition 2.6: Spatiotemporal Rips filtration spatiotemporal
• Remark 2.7: Interleaving distance between $\mathbf{Dyn}$-modules
• Definition 2.8: Order-connectivity
• Definition 2.9: Interval
• Definition 2.10: Segment