bifiltered vietoris-rips

This work investigates the fundamental limits of sparsely approximating Vietoris–Rips filtrations without any geometric assumptions. By constructing specific families of metric spaces and combining techniques from homotopy interleaving theory, combinatorial topology, and metric geometry, the authors establish the first lower bounds on approximation size within the homotopy interleaving framework: for any fixed approximation factor \( c \in [1, \sqrt{2}) \), there exist instances requiring exponentially large representations, and for any \( c \geq 1 \), the approximation size must be super-linear. These results demonstrate that geometric conditions—such as bounded doubling dimension—are essential for achieving efficient approximations and extend to several bifiltration settings, thereby ruling out the possibility of universally linear-size approximations.

This study addresses the stability of zero-dimensional persistent homology and codimension-one homology for point clouds after preprocessing—including barycentric subdivision (enrichment), iterative minimum-separation-distance sparsification, and grid alignment via coordinate discretization—motivated by topological modeling of high-dimensional environmental spaces in species distribution inference. We propose a theoretically grounded analytical framework with rigorous topological guarantees. Specifically, we establish the first exact stability bounds for persistence diagrams across distinct simplicial complexes: Vietoris–Rips, alpha, and cubical complexes. Moreover, we discover a duality identity for cubical complexes, resolving structural preservation limitations inherent in conventional approaches. Leveraging GUDHI, we implement TopoAware—a cross-language (C++/Python/R) open-source toolkit. Empirical evaluation confirms its stability and robustness under three filtration schemes.

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.

Analyzing and comparing non-hierarchical multi-scale clustering sequences remains challenging due to the lack of stable, topology-aware representations. Method: We propose Multi-scale Clustering Filtration (MCF), a stable simplicial filtration that encodes clustering partitions at arbitrary scales. We systematically introduce persistent homology to this task by constructing MCF and its equivalent nerve complex, proving that in the hierarchical case it reduces to the Vietoris–Rips filtration on an ultrametric space. Contribution/Results: Empirical validation on synthetic data demonstrates that zero- and higher-dimensional persistence diagrams derived from MCF serve as robust topological features, effectively characterizing and distinguishing diverse multi-scale clustering structures. MCF thus establishes a novel paradigm for the quantitative evaluation and comparative analysis of multi-scale clustering, enabling principled, topology-driven assessment beyond traditional metrics.

This work addresses the lack of stability for large graded persistent barcodes in finite pseudo-metric spaces. Methodologically, it introduces bifiltered persistent double homology theory: starting from the Vietoris–Rips filtration, it constructs the associated projective angular complex and defines both ordinary and double homology—yielding bifiltered persistent homology modules and barcodes. The paper establishes, for the first time, an $L^infty$-stability theorem for this bifiltered structure, rigorously proving the robustness of both modules and barcodes under perturbations of the input metric. By extending persistent homology from the conventional single-parameter (i.e., singly graded) setting to a two-parameter (bifiltered, doubly graded) framework, this work provides a novel algebraic-topological foundation and theoretical guarantee for high-dimensional topological data analysis—particularly for multi-scale and multiparameter feature extraction.

本文针对有限度量空间的持久同调问题，提出新的近似算法，通过贪婪排列等技术，在线性时间内计算稳定条形码的ε-近似值。

This study addresses the scalability limitations of multiparameter persistent homology on large-scale data by proposing the sublevel Flood bifiltration as an efficient approximation of the sublevel offset bifiltration. By extending the single-parameter Flood filtration to the biparameter setting, the proposed method integrates topological data analysis with bifiltration theory, effectively balancing topological stability and computational efficiency. The effectiveness and robustness of this approach are validated through classification tasks on both synthetic and real-world time series datasets. Ultimately, this work provides a scalable new pathway for extracting multiparameter topological features from large-scale data.

This study addresses the efficiency and structural bottlenecks arising from the inability of simplex trees in flag complexes to implicitly store attaching data, and proposes the QF-tree data structure. By labeling collapsed faces and indexing quotient vertex words, this method efficiently encodes the cells and attaching maps of quotient CW-complexes. Grounded in a theoretical proof that finite dimensions determine the entire structure, it achieves linear space complexity and local updates, significantly outperforming homotopy-equivalent cone models. Furthermore, by integrating techniques such as zigzag persistence and Trie prefix trees, this work decouples quotient maintenance from algebraic computation costs. A comprehensive function library supporting complex topological operations is also provided, validating the compact and efficient computational advantages of the proposed approach.

This study addresses the inefficiency of recomputing persistent homology from scratch when lower-star filtrations change, proposing an incremental update framework grounded in discrete Morse theory. By constructing a colex vector field compatible with a total order, the method leverages vertex transpositions to rapidly reconstruct the vector field and introduces an optimized path-counting storage and update mechanism that avoids global recomputation. This approach enables the efficient maintenance of persistent homology under dynamically changing filtrations. Open-source proof-of-concept implementations demonstrate that the proposed algorithm achieves runtime performance comparable to standard matrix reduction methods on persistent homology transformation tasks.

This work addresses the challenge of efficiently computing (persistent) discrete homology for graphs derived from high-noise, non-metric data. The authors propose a novel approach that reformulates the discrete homology problem in terms of zero differential forms and integrates an active enumeration strategy with zero-differential reduction techniques. This combination substantially improves computational efficiency for higher-order homology groups. The method achieves the first successful computation of the fourth homology group of the Greene sphere and resolves several previously unknown homology groups. When applied to noisy graph data, it significantly outperforms conventional Vietoris–Rips simplicial complex-based approaches in both speed and accuracy, yielding more precise topological features.