persistent homology

This work addresses the sensitivity of the Mapper algorithm to lens functions, cover parameters, and clustering strategies, for which no systematic evaluation framework previously existed. The authors propose the first triaxial assessment framework that comprehensively evaluates Mapper variants across three complementary dimensions: stability, cluster quality, and topological shape preservation. Experiments on synthetic data and the UCI handwritten digits dataset reveal inherent trade-offs among these dimensions, demonstrating that no single configuration achieves optimal performance across all metrics simultaneously. The study further identifies a “topological explosion” phenomenon at high resolutions, offering practical guidance for parameter selection in real-world applications and highlighting key challenges for future research in Mapper-based topological data analysis.

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 investigates the expressive power of topological descriptors—specifically Euler characteristic (EC) and persistent homology (PH)—on graph products, with focus on their behavior under color-based filtration and graph product structures. We propose *product filtering*, a novel paradigm that first fully characterizes the expressive limits of EC under color filtration. Theoretically, we prove that PH on virtual graph products captures cross-graph topological interactions inaccessible via single-graph computation. We further design the first efficient algorithm for computing persistent homology on graph products, supporting multi-scale vertex- and edge-level filtrations. Experiments demonstrate that our framework significantly outperforms conventional methods in computational efficiency, topological expressivity, and graph classification accuracy. The implementation is publicly available.

In neuroimaging research, the diversity of rfMRI brain representations—e.g., parcellation schemes and feature types—induces inconsistent individual variability estimates, severely undermining result reproducibility and cross-study integration. Method: We propose a topological similarity assessment framework based on persistent homology, specifically designed for small-sample, high-dimensional neuroimaging data. It integrates Vietoris–Rips complexes, topological bootstrapping, and hierarchical clustering, and introduces a novel prevalence-weighted Wasserstein distance to enable unbiased topological comparison across samples and heterogeneous metric spaces. Contribution/Results: We demonstrate that low-persistence but high-prevalence homological generators encode interpretable biological signals. Validation on large-scale cohorts reveals that environmental dimensions of representations—not decomposition rank—predominantly govern topological feature count and stability. The framework enables robust, unbiased clustering across diverse representations, advancing reproducible, integrative neuroimaging analysis.

To address robust clustering of unstructured point clouds, this paper proposes a novel spectral clustering method based on the global topological contribution of point pairs. The core innovation lies in constructing a simplicial complex from the point cloud and incorporating the Hodge–Laplacian operator to analyze its higher-order spectral properties, thereby capturing multi-order (beyond pairwise) topological relationships among points. This work is the first to systematically integrate Hodge–Laplacian spectral analysis into a point cloud clustering framework, synergizing sparse eigenvector computation with topological data analysis (TDA) principles to yield interpretable and noise-resilient cluster partitions. Extensive experiments on synthetic and real-world datasets demonstrate that the proposed method achieves significantly higher clustering accuracy and stability than classical spectral clustering—particularly under challenging conditions including noise corruption, non-convex cluster shapes, and multi-scale structures.

This work proposes a hierarchical clustering algorithm grounded in topological data analysis to address the challenge of identifying clusters of arbitrary shape and salient outliers without requiring prior assumptions about data distribution. The method operates under any distance metric and eschews distributional assumptions by constructing a topological hierarchy and incorporating persistence analysis to rigorously assess cluster stability and outlier significance. Experimental evaluations on real-world datasets from domains such as image analysis, healthcare, and economics demonstrate that the algorithm yields semantically coherent and robust clustering results even in complex scenarios where conventional approaches fail, substantially enhancing the detection of non-convex structures and critical anomalous points.

This study addresses the automatic and robust identification of genuine topological signal points in persistence diagrams (PDs) to enhance the interpretability and practical applicability of topological data analysis. To this end, we propose TUN—the first multimodal neural network designed for semantic understanding of PDs—integrating enhanced geometric/topological descriptors, self-attention mechanisms, PointNet-style point cloud encoding, and a learnable multimodal feature fusion module for end-to-end salient point detection. To handle PDs’ inherent sparsity and extreme class imbalance, we introduce specialized preprocessing and weighted training strategies. Evaluated on multiple standard benchmarks, TUN significantly outperforms both classical heuristic and existing deep learning methods in detection accuracy and robustness. Our approach establishes a generalizable technical paradigm for automated PD interpretation, advancing the deployment of topological features in real-world applications.

In unsupervised representation learning, embedding quality assessment has long been constrained by strong assumptions—such as linear separability or covariance structure—due to the absence of ground-truth labels. To address this, we propose *Persistence*, the first unsupervised, topology-aware evaluation metric grounded in persistent homology. It quantifies multi-scale geometric structure and topological richness of embedding spaces, enabling a unified characterization of global, nonlinear patterns without requiring labels or model-specific assumptions—thereby overcoming theoretical limitations of conventional metrics. Empirically validated across diverse domains, Persistence achieves the highest correlation with downstream task performance (average Pearson *r* = 0.89), significantly outperforming existing unsupervised evaluation methods. It effectively supports model selection and hyperparameter optimization.

This work addresses the lack of topological information in existing 3D shape datasets, which hinders joint geometric and topological learning. To bridge this gap, the authors introduce the first topologically enriched versions of the ModelNet40 and ShapeNet benchmark datasets by incorporating persistent homology features. They propose TopoGAT, an end-to-end graph attention network architecture that employs a learnable mechanism to select the most salient persistence diagram points, thereby automatically extracting discriminative topological features. Evaluated on 3D point cloud classification and part segmentation tasks, TopoGAT significantly outperforms conventional handcrafted topological feature methods, demonstrating the critical role of topological information in enhancing both model performance and robustness.

Existing medical image classification methods either neglect anatomical structures—such as topological invariants—or capture only simplistic topological features via single-parameter persistence. To address this, we propose the first end-to-end multi-scale, multi-filter topologically guided framework: it computes multi-resolution persistent homology on cubical complexes, integrates multi-scale persistence diagrams using the vineyard algorithm, and employs a cross-attention network to fuse multi-filter topological features; the framework is plug-and-play compatible with CNN or Transformer backbones. Evaluated on three public medical image datasets, our method significantly outperforms strong baselines and state-of-the-art models. Results demonstrate that multi-scale, multi-filter topological representations substantially enhance classification robustness, interpretability, and generalization capability.