topology-based filtering

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.

This work addresses the low efficiency and poor structural preservation inherent in prototype selection for large-scale datasets. We propose TPS, a topology-aware prototype selection framework grounded in topological data analysis (TDA). TPS leverages persistent homology to characterize the intrinsic geometry and connectivity structure of data, enabling adaptive identification of topologically salient samples as prototypes; it further supports parallel implementation. Compared with conventional methods, TPS achieves substantial data compression—retaining only 5–15% of samples on multiple synthetic and real-world benchmarks—while maintaining or improving classification accuracy by 1.2–3.8 percentage points. The approach also exhibits strong interpretability and robustness. Its core innovation lies in the first systematic integration of TDA’s structural awareness into prototype selection, thereby unifying computational efficiency, structural fidelity, and interpretability.

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 critical challenges of robustness, generalization, and interpretability faced by artificial intelligence models in high-stakes domains such as defense, particularly when handling diverse data modalities including images, time series, and graph-structured data. To this end, the study proposes a unified framework that, for the first time, systematically integrates topological neural networks, topological data analysis (TDA), and Bayesian deep learning. This integration enables simultaneous capture of complex intrinsic geometric structures within data and principled quantification of model uncertainty. The resulting approach demonstrates significant performance improvements across a range of tasks—including image, video, and audio recognition, fraud detection, and graph link prediction—thereby delivering more reliable, interpretable, and generalizable AI solutions for mission-critical applications.

Existing image segmentation methods predominantly rely on pixel-wise losses (e.g., Dice), neglecting topological consistency; mainstream topology-aware approaches either lack rigorous theoretical guarantees or suffer from high computational cost and poor generalizability. Method: We propose the first differentiable, lightweight, and formally guaranteed topology-preserving framework: (i) we introduce and optimize a strict homotopy equivalence metric; (ii) we construct a differentiable component graph based on connected components, enabling local neighborhood-sensitive topological modeling and loss computation; (iii) we employ graph neural networks for feature aggregation and homotopy classification to enforce homotopy equivalence between predictions and ground truth. Contribution/Results: Our method achieves state-of-the-art performance on diverse multi-class medical and natural image segmentation benchmarks. It significantly improves topological accuracy and accelerates topological loss computation by 5× compared to persistent homology–based methods.

This work addresses the challenge in non-negative matrix factorization (NMF) of simultaneously achieving interpretable basis functions and desirable topological structure, a limitation often exacerbated by reliance on discrete formulations or thresholding. To overcome this, the study introduces persistent homology into NMF for the first time, proposing a threshold-free, continuously optimized topological regularization framework. By embedding a topological loss term derived from persistent homology directly into the objective function, the method guides the learning of non-negative basis functions that exhibit specific topological features—such as spatial coherence, periodicity, or clique-like structures. The approach is unified across diverse data modalities, including images, time series, and graph signals, significantly enhancing both the structural plausibility and interpretability of the resulting decompositions.

This work addresses the limitations of traditional graph signal processing, which is confined to node-level signals and thus unable to capture higher-order interactions inherent in complex systems. By leveraging simplicial complexes and combinatorial Hodge Laplacians, the study extends signal processing to higher-dimensional topological structures such as edges and triangles. It introduces a method for constructing higher-order signals from lagged node observations and develops a corresponding theory of topological Fourier transforms and filtering. Applied to brain imaging data, the proposed framework successfully uncovers nontrivial higher-order interaction patterns among sets of brain regions that are invisible to conventional approaches, thereby establishing a theoretical and practical bridge for higher-order topological signal processing.

This study investigates whether the decision regions of deep image classifiers are simply connected—meaning any closed loop within such a region can be continuously contracted to a point without leaving the region. To address this, the authors propose an iterative quadrilateral mesh-filling method that constructs a finite-resolution surface bounded by a given loop while preserving label consistency, and they employ Coons surfaces to quantify deviations from standard geometric interpolation. Large-scale experiments across multiple state-of-the-art models provide strong empirical evidence that decision regions are indeed simply connected. This work offers the first empirical support for the simple connectivity of deep neural network decision regions, advancing beyond prior studies limited to path connectivity and deepening the understanding of the geometric and topological properties of decision boundaries.

This study addresses the challenge of accurately predicting professional tennis match outcomes in the absence of conventional player rankings. Leveraging ATP singles match data from 2000 to 2025, the authors construct a competitive network among players and introduce the down-star filtration—a novel application in tennis prediction—while systematically evaluating four topological summary methods (VAB, HNAV, HWNAV, and OW-HNPV). By integrating persistent homology, refined band-depth analysis, centrality measures, an enhanced Katz similarity index, and time-weighted edges, their purely topological model achieves a prediction accuracy of 63.56% without any ranking information. A hybrid model incorporating additional features further improves performance to 66.2% accuracy (AUC = 0.719), demonstrating that network topological features provide significant complementary value for match outcome prediction.

This work addresses the formalization and verification of robustness in machine learning classifiers against adversarial perturbations. It introduces a novel logical framework that integrates topological semantics with modal logic, grounded in S4 topological spaces, wherein robustness is interpreted as the persistence of local truth. The core contributions include the introduction of a robustness modality and a new robustness-sensitive conditional connective to precisely capture inclusion relations among robust regions and classification conditions. Furthermore, the paper proposes a constructive method for generating minimal robust models and establishes a complete axiomatization for the proposed logic. It also provides a formal toolchain that automatically synthesizes models from robustness constraints, enabling structured analysis, interpretation, and modeling of classifier robustness.