Spatiotemporal Persistence Curves

• Spatiotemporal persistence curves are mathematical constructs that quantify evolving topological features across space and time using generalized rank invariants.
• They leverage extended zigzag modules and multiparameter persistence to analyze time series, spatial signals, and dynamic systems with robust stability.
• These curves enable practical applications in statistics and machine learning by supporting vectorization, averaging, and hypothesis testing of complex topological data.

Spatiotemporal persistence curves are mathematical objects arising in topological data analysis that quantify the evolution of topological features—such as connected components, cycles, and voids—simultaneously across space and time. They are generalizations of classical persistence landscapes, grounded in persistent homology, but extend their domain from one-parameter filtrations to filtrations indexed by more complex posets that encode both spatial and temporal structure. This generalization is accomplished through the construction of extended zigzag modules and the assignment of generalized rank invariants, yielding Banach space–valued invariants that exhibit stability under adapted interleaving distances. These structures enable rigorous statistical and algorithmic analysis on time series data, spatial-temporal signals, and dynamical systems, supporting applications in machine learning pipelines and statistical inference.

1. Generalized Rank Invariant and Interval-Decomposable Modules

A foundational concept for spatiotemporal persistence landscapes is the generalized rank invariant, introduced in the context of persistent modules over arbitrary posets. Standard persistence diagrams are complete invariants for modules indexed by totally ordered sets. However, when working over a parameter space such as the product of space and time (which forms a partially ordered set), classical diagrams become insufficient.

The generalized rank invariant records the rank of the restriction of a module to every interval $I$ in the poset. For an interval-decomposable module $M$, the rank over $I$ is computed via the canonical map from the limit to the colimit of $M$ restricted to $I$. In precise terms, for such modules, the rank over $I$ corresponds to the count of interval summands—interval modules—whose domains contain $I$. This yields a complete invariant (as proved in Emoli 2021 and in related works), which then serves as the algebraic basis for defining and computing spatiotemporal persistence landscapes.

2. Extended Zigzag Modules: Multiparameter and Zigzag Persistence

Spatiotemporal persistence landscapes require filtrations indexed not by a linear order but by a poset encoding simultaneous progression in space and time. Such filtrations are naturally associated with extended zigzag modules. These combine the flexibility of multiparameter persistence (to handle more than one geometric direction) with zigzag persistence (to accommodate both additions and deletions in the filtration sequence).

The construction involves tracking modules along paths in the poset determined by pairs of boundaries (called lower and upper fences), which define the intervals for calculating the generalized rank. The zigzag structure allows for crossing back and forth in parameter space, encoding persistence of features that may both emerge and disappear at non-monotonic locations in space-time. This module-theoretic approach is required to capture phenomena in time series and dynamic networks where classical monotone filtrations are inadequate.

3. Definition and Properties of Spatiotemporal Persistence Landscapes

Spatiotemporal persistence landscapes are defined from these extended zigzag modules using the generalized rank invariant. For each position $x$ (in parameter space, e.g., space-time) and each scale parameter $r$, the landscape function assigns the maximal $r$ such that the generalized rank is at least $k$. Formally, these landscapes are families of functions taking values in Lebesgue spaces, specifically:

$$\lambda_k(x) = \sup \{ r : \operatorname{rank}_M(B(x, r)) \ge k \},$$

where $B(x, r)$ denotes the “ball” or interval in the poset centered at $x$ with radius $r$, and $\operatorname{rank}_M$ is the generalized rank as introduced above. Each landscape encodes, for each $k$, the persistence of the $k$-th most persistent feature within the spatiotemporal domain.

4. Stability and Distance Structure

A distinguishing property of spatiotemporal persistence landscapes is their stability: small perturbations in the input data (filtration or topological signal) induce small changes in the landscape (measured via an appropriate norm). The stability result is formulated with respect to an adapted interleaving distance designed for extended zigzag modules. Specifically, under suitable finiteness assumptions, the space of spatiotemporal persistence landscapes can be endowed with a norm inherited from integration over Lebesgue spaces, and the landscape distance between two signals is upper bounded by the interleaving distance between the underlying modules.

This stability parallels foundational results for classical persistence diagrams but is technically more delicate due to the complex structure of the indexing poset. It ensures that statistical and machine learning methods using such landscapes as features are robust to data noise and sampling variability.

5. Statistical and Algorithmic Applications

Because spatiotemporal persistence landscapes are Banach space–valued invariants (that is, they live in complete normed vector spaces), they can be subjected to statistical operations such as averaging, principal component analysis, and hypothesis testing. Landscapes can be vectorized for input into machine learning algorithms, supporting supervised classification, clustering, and regression tasks on time-ordered or spatiotemporal data.

This vectorization leverages the theoretical completeness and stability of the rank invariant, ensuring that the resultant features adequately summarize topological structure and persistence across both space and time. Applications include time series analysis, activity detection in sensor networks, dynamic system modeling, and image sequence classification.

6. Theoretical Connections and Comparisons

The theoretical architecture of spatiotemporal persistence landscapes is built on the rank invariant theory developed in Emoli (2021), as well as categorical insights involving limits, colimits, and Kan extensions. Multiple versions of Emoli’s work—journal and preprint—offer refined definitions and computational frameworks for generalized persistence diagrams over arbitrary posets. The construction of spatiotemporal persistence landscapes utilizes similar categorical arguments to establish equivalence between the generalized rank computed by restriction to intervals and the “boundary” approach via lower and upper fences.

This ensures the landscape is both computable and interpretable within the framework of extended zigzag modules, with stability results directly analogous to those established for generalized persistence diagrams. The cross-compatibility between the two theories is essential for the practical computation and application of spatiotemporal invariants.

7. Significance for Topological Data Analysis and Future Research

Spatiotemporal persistence landscapes bridge a critical gap in topological data analysis, enabling the quantification and visualization of features that persist non-trivially across both spatial and temporal domains. By combining multiparameter and zigzag persistence, they extend the set of tools available for analyzing time series, dynamical systems, and complex evolving shapes. Their amenability to statistical and machine learning workflows enhances their practical utility.

Anticipated future directions encompass efficient algorithms for landscape computation over large parameter spaces, further development of statistical inference frameworks for Banach space–valued invariants, and applications to emerging domains such as network neuroscience, spatiotemporal epidemiology, and automated discovery in dynamic environments. The stability and completeness of these invariants ensure that subsequent analyses are theoretically sound and empirically robust.