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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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EMP: Effective Multidimensional Persistence for Graph Representation Learning
Topological data analysis (TDA) is gaining prominence across a wide spectrum of machine learning tasks that spans from manifold learning to graph classification. A pivotal technique within TDA is persistent homology (PH), which furnishes an exclusive topological imprint of data by tracing the evolution of latent structures as a scale parameter changes. Present PH tools are confined to analyzing data through a single filter parameter. However, many scenarios necessitate the consideration of multiple relevant parameters to attain finer insights into the data. We address this issue by introducing the Effective Multidimensional Persistence (EMP) framework. This framework empowers the exploration of data by simultaneously varying multiple scale parameters. The framework integrates descriptor functions into the analysis process, yielding a highly expressive data summary. It seamlessly integrates established single PH summaries into multidimensional counterparts like EMP Landscapes, Silhouettes, Images, and Surfaces. These summaries represent data’s multidimensional aspects as matrices and arrays, aligning effectively with diverse ML models. We provide theoretical guarantees and stability proofs for EMP summaries. We demonstrate EMP’s utility in graph classification tasks, showing its effectiveness. Results reveal EMP enhances various single PH descriptors, outperforming cutting-edge methods on multiple benchmark datasets.
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- Award ID(s):
- 2220613
- PAR ID:
- 10514900
- Editor(s):
- Lawrence, Neil
- Publisher / Repository:
- PMLR
- Date Published:
- Journal Name:
- Proceedings of the Second Learning on Graphs Conference
- ISSN:
- 2640-3498
- Subject(s) / Keyword(s):
- Multiparameter Persistence, topological data analysis, graph representation learning
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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