persistence diagram matching

In supervised learning based on persistent homology, computing persistence diagrams is computationally expensive, and conventional full matrix reduction often discards essential topological information from the original data. To address this, we propose a novel paradigm that directly extracts topological feature vectors from the **unreduced boundary matrix**, bypassing costly reduction while preserving richer algebraic topological structure. Our method is grounded in persistent homology theory and introduces a differentiable, scalable feature mapping mechanism. We conduct systematic evaluations across diverse datasets and tasks—including classification and regression. Experiments demonstrate that our approach matches or surpasses standard reduced-persistence baselines in predictive performance, while substantially reducing computational complexity. These results empirically validate our core claim: strong discriminative topological features can be obtained *without full matrix reduction*. The work thus establishes a new pathway toward efficient topological machine learning.

This work addresses the limitation of steady and ranging persistence—originally defined only for graphs—to generalized combinatorial structures such as hypergraphs. We develop a unified, category-theoretic axiomatic framework and, for the first time, formally define and characterize necessary and sufficient conditions for induced balanced persistence. Building on this foundation, we design a hypergraph filtration method, yielding the first computable instance of balanced persistence. Theoretically, we establish stability guarantees under the interleaving distance. Empirically, we demonstrate the method’s effectiveness and robustness on diverse hypergraph datasets. All code and experiments are fully open-sourced and reproducible. This work provides a novel paradigm for extending topological data analysis to higher-order relational modeling.

The matching distance for two-parameter persistence modules over ℝ² is notoriously difficult to compute exactly. Method: This paper introduces the first implementable exact algorithm for computing the 2D matching distance. We establish, for the first time, an explicit geometric connection between the matching distance and critical values in the parameter space—proving that any optimal matching line must intersect a critical structure. Leveraging this insight, we construct a piecewise-linear geometric framework that integrates critical-point tracking with multiparameter persistence module theory to enable efficient computation. Contributions: (1) A geometric characterization of the matching distance with formal interpretability guarantees; (2) An exact, efficient, and implementable algorithm for the 2D case; (3) The first practical, exact computational framework enabling multiparameter persistent homology to be deployed in real-world data analysis.

This work addresses the computational inefficiency of computing the 1-Wasserstein distance between large-scale persistent diagrams (PDs). To this end, we propose PDoptFlow—the first open-source framework enabling near-linear-time, high-accuracy approximation. Methodologically, we introduce the first tight lower bound for PD distances by integrating Well-Separated Pair Decomposition (WSPD) with the relaxed Word Mover’s Distance lower bound. We further propose a dual sparsification strategy—applied to both nodes and edges—to formulate the optimal transport problem as a sparse minimum-cost flow network. Finally, we design a GPU–multi-core co-parallel solver for efficient computation. Experiments demonstrate that PDoptFlow achieves <1% relative error while outperforming state-of-the-art methods by one to two orders of magnitude in runtime, enabling scalable 1-Wasserstein distance computation for PDs containing up to millions of points.

This work addresses the challenge of geometrically consistent matching of topological features—particularly persistent homology cycles—across disparate data systems. We propose Topological Optimal Transport (TpOT), the first framework that deeply integrates optimal transport with persistent homology. TpOT constructs a measure-topological network and defines a differentiable, geometry-aware topological-geometric joint distance within its non-negatively curved geodesic metric space, leveraging hypergraph optimal transport, measure theory, and Riemannian-geometric optimization of transport plans. On point cloud data, TpOT significantly reduces topological distortion while producing geometrically plausible and interpretable cycle-level correspondences. Theoretically, we prove that the proposed distance satisfies all metric axioms. TpOT establishes the first differentiable matching paradigm for topological data analysis that simultaneously ensures geometric fidelity and topological faithfulness.

This work addresses a critical limitation in existing vectorization and kernel methods for topological data analysis, which neglect the rank-induced inclusion relations among persistence intervals, thereby distorting structural information and reducing interpretability. To overcome this, the authors propose high-order persistence diagrams that explicitly capture structural dependencies by recursively modeling interval inclusion relationships. They further introduce, for the first time, harmonic analysis and the zeta transform to implicitly aggregate high-order diagrams in the spectral domain, reducing computational complexity from quadratic to nearly linear. Experimental results demonstrate that the proposed method significantly outperforms explicit aggregation strategies on random network models, achieving superior efficiency and scalability while preserving structural fidelity.

This work addresses the ongoing challenge of effectively incorporating topological priors into optimization problems. It proposes a systematic framework for topological optimization grounded in persistent homology, which unifies gradient-based optimization with a differentiable topological regularization loss to enable end-to-end learning of topological features. Providing a comprehensive survey of theoretical and algorithmic advances over the past decade, the paper offers—for the first time—an accessible, unified introduction tailored to mathematicians and data scientists new to the field. Accompanied by an open-source library, this contribution aims to lower entry barriers and foster broader adoption of topological data analysis in machine learning and data science communities.

Traditional persistence diagrams struggle to capture the interactive topological relationships between point clouds and lack cross-structural modeling capacity. This work establishes, for the first time, the existence and statistical foundations of cross-persistence diagram densities and introduces an end-to-end framework that integrates topological data analysis, statistical learning, and deep learning to directly predict these densities from point cloud coordinates and distance matrices. A novel noise-augmentation mechanism is innovatively incorporated to enhance the discriminative power of point clouds, significantly extending the applicability of topological data analysis in cross-structural settings. Experiments demonstrate that the proposed method achieves state-of-the-art performance in both density prediction and point cloud discrimination across multiple datasets, while also showing promising potential in geometric analyses of time series and AI-generated text.

本文提出了一种名为Sierpiński-Knopp Wasserstein距离的快速度量方法，用于解决持久图间的比较问题，通过空间填充曲线实现高效匹配，并在多个实验中展示了其速度和准确性。

Existing vectorization methods for expected persistence diagrams (EPDs) rely on predefined smoothing transformations, which struggle to adaptively capture the distribution of topological features. This work proposes a Voronoi diagram–based histogram vectorization approach that achieves adaptive discretization of EPDs through spatial partitioning and counting, eliminating the need for explicit smoothing functions. To the best of our knowledge, this is the first method to incorporate Voronoi histograms into EPD representations, offering theoretical guarantees of stability and information preservation under the Wasserstein metric under certain conditions. Experimental results demonstrate that the proposed representation effectively captures essential topological features and outperforms conventional vectorization techniques in classification and dimensionality reduction tasks on real-world datasets.