Dynamical Persistent Homology via Wasserstein Gradient Flow
Minghua Wang, Jinhui Xu
TL;DR
This paper presents new algorithms that translate changes in persistence diagrams into data space modifications, enabling dynamic data analysis guided by Wasserstein gradient flows in topological data analysis.
Contribution
It introduces methodologies for adapting data based on the evolution of persistence diagrams along Wasserstein gradient flows, bridging topological summaries and data manipulation.
Findings
Algorithms for translating persistence diagram variations into data modifications
Enhanced understanding of data dynamics through topological methods
Potential applications in data analysis and interpretation
Abstract
In this study, we introduce novel methodologies designed to adapt original data in response to the dynamics of persistence diagrams along Wasserstein gradient flows. Our research focuses on the development of algorithms that translate variations in persistence diagrams back into the data space. This advancement enables direct manipulation of the data, guided by observed changes in persistence diagrams, offering a powerful tool for data analysis and interpretation in the context of topological data analysis.
Peer Reviews
Decision·ICLR 2026 Conference Withdrawn Submission
This paper proposes an algorithm for solving the inverse problem of persistent homology based on the _p_-Wasserstein distance and provides a proof of its convergence.
The concerns about this paper can be summarized in two main points: - First, it is unclear whether the content of this paper fits within the scope of the conference. This venue focuses on AI, machine learning, and representation learning, and the paper should therefore demonstrate its relevance or effectiveness in these contexts. While the proposed method may indeed have potential applications in AI or machine learning tasks, the current presentation does not explicitly describe such connection
- Mathematically sound, and conceptually simple. - Code is available, experiments are reproducible via simple notebooks.
- Theoretical guarantees. The (continuous) theory relies on well established optimal transport theory, but the resulting geodesics are discretized. Are there some bounds on the error? - Experiements. - Synthetic datasets only. - Can these results be compared to other methods (with other losses)? - The experiements are on very small datsets, how does this scale to larger datsets? - This approach is based on several approximations schemes. Are there empirical validations on known
1. Novelty. The paper frames an inverse problem for TDA by evolving persistence diagrams in Wasserstein space and pulling those dynamics back through differentiable persistent homology, bridging optimal transport and persistence optimization. 2. It provides two complementary realizations: McCann interpolation toward a single target diagram with OT couplings and filtration updates, and a JKO energy-based scheme without a target, with clear algorithmic details and discussion of achievability of i
1. While the idea is very nice, empirical evidence is limited to qualitative toy demos with circles, without quantitative metrics, baselines, or ablations to assess robustness or superiority. 2. No guarantee that each desired diagram step is realizable by filtration updates; the method explicitly carries forward unattained targets ($Y^{(k)}$), highlighting an achievability gap. 3. Scalability and scope are constrained: current implementation is CPU bound, focuses only on standard persistence d
- The paper covers the background from both optimal transport and TDA sides, which is a challenge in itself. - Overall, the paper is well-structured.
1. Lack of originality 2. Missing motivation for Alg. 1 3. (assuming the answer to q. 1 is yes) suspiciously non-standard treatment of the Wasserstein distance between PDs as optimal transport problem. 1. The cited paper by Nigmetov and Morozov (by the way, as far as I know, the idea of modifying the shape of input by backpropagating through persistence diagrams first appeared in "A topology layer for machine learning" by Brüel-Gabrielsson et al., which probably should be cited) contains the ma
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Taxonomy
TopicsTopological and Geometric Data Analysis · Advanced Neuroimaging Techniques and Applications · Homotopy and Cohomology in Algebraic Topology