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1. Time-Aware Knowledge Representations of Dynamic Objects with Multidimensional Persistence

   https://doi.org/10.1609/aaai.v38i10.29051  

   Coskunuzer, Baris; Segovia-Dominguez, Ignacio; Chen, Yuzhou; Gel, Yulia R (March 2024, Proceedings of the AAAI Conference on Artificial Intelligence)

   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 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. 

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2. Comparing Distance Metrics on Vectorized Persistence Summaries

   Fasy, Brittany Terese; Qin, Yu; Summa, Brian; Wenk, Carola (December 2020, Topological Data Analysis and Beyond Workshop at the 34th Conference on Neural Information Processing Systems (NeurIPS 2020))

   The persistence diagram (PD) is an important tool in topological data analysis for encoding an abstract representation of the homology of a shape at different scales. Different vectorizations of PD summary are commonly used in machine learning applications, however distances between vectorized persistence summaries may differ greatly from the distances between the original PDs. Surprisingly, no research has been carried out in this area before. In this work we compare distances between PDs and between different commonly used vectorizations. Our results give new insights into comparing vectorized persistence summaries and can be used to design better feature-based learning models based on PDs 

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3. Topology-aware robust representation balancing for estimating causal effects

   Farzam, Amirhossein; Aloui, Ahmed; Tarokh, Vahid; Sapiro, Guillermo (July 2025, NeurIPS 2025 High-dimensional Learning Dynamics Workshop)

   Representation learning in high-dimensional spaces faces significant robustness challenges with noisy inputs, particularly with heavy-tailed noise. Arguing that topological data analysis (TDA) offers a solution, we leverage TDA to enhance representation stability in neural networks. Our theoretical analysis establishes conditions under which incorporating topological summaries improves robustness to input noise, especially for heavy-tailed distributions. Extending these results to representation-balancing methods used in causal inference, we propose the *Topology-Aware Treatment Effect Estimation* (TATEE) framework, through which we demonstrate how topological awareness can lead to learning more robust representations. A key advantage of this approach is that it requires no ground-truth or validation data, making it suitable for observational settings common in causal inference. The method remains computationally efficient with overhead scaling linearly with data size while staying constant in input dimension. Through extensive experiments with -stable noise distributions, we validate our theoretical results, demonstrating that TATEE consistently outperforms existing methods across noise regimes. This work extends stability properties of topological summaries to representation learning via a tractable framework scalable for high-dimensional inputs, providing insights into how it can enhance robustness, with applications extending to domains facing challenges with noisy data, such as causal inference. 

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4. Persistence Images: A Stable Vector Representation of Persistent Homology

   Adams, H; Emerson, T; Kirby, M; Neville, R; Peterson, C; Shipman, P; Chepushtanova, S; Hanson, E.; Motta, F; Ziegelmeier, L (January 2017, Journal of machine learning research)

   Many data sets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a data set. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs. 

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5. Topology-Informed Pre-training of Graph Neural Networks

   https://doi.org/10.1109/ICASSP49660.2025.10890271  

   Liang, Peiyu; Gel, Yulia R; Chen, Yuzhou (April 2025, IEEE)

   Pre-training has emerged as a dominant paradigm in graph representation learning to address data scarcity and generalization challenges. The majority of existing methods primarily focus on refining fine-tuning and prompting techniques to extract information from pre-trained models. However, the effectiveness of these approaches is contingent upon the quality of the pre-trained knowledge (i.e., latent representations). Inspired by the recent success in topological representation learning, we propose a novel pre-training strategy to capture and learn topological information of graphs. The key to the success of our strategy is to pre-train expressive Graph Neural Networks (GNNs) at the levels of individual nodes while accounting for the key topological characteristics of a graph so that GNNs become sufficiently powerful to effectively encode input graph information. The proposed model is designed to be seamlessly integrated with various downstream graph representation learning tasks. 

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