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Time-Aware Knowledge Representations of Dynamic Objects with Multidimensional Persistence

Baris Coskunuzer, Ignacio Segovia-Dominguez, Yuzhou Chen, Yulia R. Gel

TL;DR

This work tackles learning from time-evolving data by introducing Temporal MultiPersistence (TMP), which combines multi-parameter persistence with zigzag persistence along the time axis to produce time-aware, multidimensional topological fingerprints. TMP vectorizations extend standard persistence representations into higher dimensions, yielding ML-friendly inputs such as tensors that capture evolution across time and other filtering dimensions. The authors prove stability guarantees for TMP vectorizations, propose TMP-Nets that integrate adaptive GCNs with CNN-based TMP feature learning and GRU forecasting, and demonstrate strong performance and efficiency advantages on traffic, Ethereum token networks, and ECG data, especially under limited data. Overall, TMP provides a scalable, theory-grounded framework that unifies topological summaries with deep learning for robust time-aware learning on dynamic graphs and time series.

Abstract

Learning time-evolving objects such as multivariate time series and dynamic networks requires the development of novel knowledge representation mechanisms and neural network architectures, which allow for capturing implicit time-dependent information contained in the data. Such information is typically not directly observed but plays a key role in the learning task performance. In turn, lack of time dimension in knowledge encoding mechanisms for time-dependent data leads to frequent model updates, poor learning performance, and, as a result, subpar decision-making. Here we propose a new approach to a time-aware knowledge representation mechanism that notably focuses on implicit time-dependent topological information along multiple geometric dimensions. In particular, we propose a new approach, named \textit{Temporal MultiPersistence} (TMP), which produces multidimensional topological fingerprints of the data by using the existing single parameter topological summaries. The main idea behind TMP is to merge the two newest directions in topological representation learning, that is, multi-persistence which simultaneously describes data shape evolution along multiple key parameters, and zigzag persistence to enable us to extract the most salient data shape information over time. We derive theoretical guarantees of TMP vectorizations and show its utility, in application to forecasting on benchmark traffic flow, Ethereum blockchain, and electrocardiogram datasets, demonstrating the competitive performance, especially, in scenarios of limited data records. In addition, our TMP method improves the computational efficiency of the state-of-the-art multipersistence summaries up to 59.5 times.

Time-Aware Knowledge Representations of Dynamic Objects with Multidimensional Persistence

TL;DR

This work tackles learning from time-evolving data by introducing Temporal MultiPersistence (TMP), which combines multi-parameter persistence with zigzag persistence along the time axis to produce time-aware, multidimensional topological fingerprints. TMP vectorizations extend standard persistence representations into higher dimensions, yielding ML-friendly inputs such as tensors that capture evolution across time and other filtering dimensions. The authors prove stability guarantees for TMP vectorizations, propose TMP-Nets that integrate adaptive GCNs with CNN-based TMP feature learning and GRU forecasting, and demonstrate strong performance and efficiency advantages on traffic, Ethereum token networks, and ECG data, especially under limited data. Overall, TMP provides a scalable, theory-grounded framework that unifies topological summaries with deep learning for robust time-aware learning on dynamic graphs and time series.

Abstract

Learning time-evolving objects such as multivariate time series and dynamic networks requires the development of novel knowledge representation mechanisms and neural network architectures, which allow for capturing implicit time-dependent information contained in the data. Such information is typically not directly observed but plays a key role in the learning task performance. In turn, lack of time dimension in knowledge encoding mechanisms for time-dependent data leads to frequent model updates, poor learning performance, and, as a result, subpar decision-making. Here we propose a new approach to a time-aware knowledge representation mechanism that notably focuses on implicit time-dependent topological information along multiple geometric dimensions. In particular, we propose a new approach, named \textit{Temporal MultiPersistence} (TMP), which produces multidimensional topological fingerprints of the data by using the existing single parameter topological summaries. The main idea behind TMP is to merge the two newest directions in topological representation learning, that is, multi-persistence which simultaneously describes data shape evolution along multiple key parameters, and zigzag persistence to enable us to extract the most salient data shape information over time. We derive theoretical guarantees of TMP vectorizations and show its utility, in application to forecasting on benchmark traffic flow, Ethereum blockchain, and electrocardiogram datasets, demonstrating the competitive performance, especially, in scenarios of limited data records. In addition, our TMP method improves the computational efficiency of the state-of-the-art multipersistence summaries up to 59.5 times.
Paper Structure (49 sections, 1 theorem, 17 equations, 2 figures, 12 tables)

This paper contains 49 sections, 1 theorem, 17 equations, 2 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Time Series Forecasting
  4. Graph Convolutional Networks
  5. Multiparameter Persistence
  6. Background
  7. Background on Persistent Homology
  8. Multidimensional Persistence
  9. TMP Vectorizations
  10. Single Persistence Vectorizations
  11. TMP Vectorizations
  12. Examples of TMP Vectorizations
  13. TMP Landscapes
  14. TMP Persistence Images
  15. Stability of TMP Vectorizations
  16. ...and 34 more sections

Key Result

Theorem 1

Let $\varphi$ be a stable vectorization for single parameter PDs. Then, the induced TMP Vectorization $\mathbf{M}_\varphi$ is also stable, i.e. With the notation above, there exists $\widehat{C}_\varphi>0$ such that for any pair of time-aware network sequences $\widetilde{\mathcal{G}}$ and $\widetil

Figures (2)

  • Figure 1: TMP outline. Given $\widetilde{\mathcal{G}}=\{\mathcal{G}_1, \mathcal{G}_2, \dots, \mathcal{G}_{T}\}$ with time-index $t=1,\ldots, T$ (1st row) we apply a bifiltration on node/edge-features at $t$, i.e. $\{\mathcal{G}_{t}^{ij}\}$ for $1\leq i\leq m$ and $1\leq j\leq n$ (2nd row). The sequence of subgraphs $\{\mathcal{G}^{i_0j_0}_1, \mathcal{G}_2^{i_0j_0}, \dots, \mathcal{G}_{T}^{i_0j_0}\}$, at fixed $i_0,j_0$ is the input into the zigzag persistence method to produce a zigzag persistence barcode (3rd row). Then, $\vec{\varphi}(\widetilde{\mathcal{G}}^{i_0j_0})$ is the corresponding vectorization for zigzag PD $ZPD_k(\widetilde{\mathcal{G}}^{i_0j_0})$ of $k-$dim feature (4th row).
  • Figure 2: Multidimensional persistence on a graph network (original graph: left). Black numbers denote the degree values of each node whilst red numbers show the edge weights of the network. Hence, shape properties are computed on two filtering functions (i.e., degree and edge weight). While each row filters by degree, each column filters the corresponding subgraph using its edge weights. For each cell, lower left corners represent the corresponding threshold values. For each cell, $\mathcal{B}_{0}$ and $\mathcal{B}_{1}$ represent the corresponding Betti numbers.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Remark 2: Stability with respect to Matching Distance