topological preservation evaluation

Conventional entropy-based diversity metrics fail to capture the intrinsic geometric and topological structure of high-dimensional training data, limiting their ability to assess data quality and predict model performance. Method: This work introduces persistent homology—a tool from topological data analysis—to quantify structural properties such as connected components, holes, and higher-order voids, thereby jointly characterizing data richness and redundancy through both topological and geometric lenses. Contribution/Results: Experiments demonstrate that the proposed topological metrics exhibit strong correlation with model generalization performance and serve as effective predictors of data quality. The framework enables principled data selection and efficient training, offering a novel paradigm for dataset curation. By leveraging topological signatures, it enhances training efficiency and robustness of AI systems without requiring model retraining or architectural modification. The approach is broadly applicable across domains where data geometry and topology critically influence learning dynamics.

This work addresses a critical limitation in existing dimensionality reduction quality metrics, which predominantly focus on pairwise relationships while overlooking structural distortions in empty regions—areas that often define key visual layouts in scatterplots. To bridge this gap, the paper introduces the Gap Index (GI), the first metric specifically designed to quantify geometric distortion in low-dimensional empty regions. GI leverages Delaunay triangulation to construct “empty triangles” in the embedding space and measures local spatial deformation by comparing their geometry with corresponding structures in the high-dimensional space. The index can be aggregated into a scalar quality score or used to visualize regional distortion patterns. Computationally efficient and highly sensitive to visually salient fine-scale structural changes, GI demonstrably outperforms conventional metrics in capturing such subtle yet perceptually important distortions.

Conventional dimensionality reduction (DR) evaluation suffers from systematic bias due to the frequent adoption of highly correlated metrics, leading to overemphasis on specific structural properties. Method: We propose an empirically grounded metric redundancy reduction framework: first computing Pearson correlation matrices across diverse datasets and DR algorithms; then applying clustering to identify functionally redundant metric groups; and finally retaining only the most representative metric per group—replacing subjective, intent-driven metric selection with objective, behavior-based clustering. Contribution/Results: Our approach significantly improves cross-dataset and cross-algorithm stability of DR evaluations, effectively mitigating structural biases inherent in traditional assessment protocols. Experimental validation demonstrates enhanced reproducibility and generalizability, establishing a principled, data-driven framework for fair and robust comparative evaluation of DR methods.

This paper addresses the challenge of jointly modeling geometric proximity and topological features in topological data analysis (TDA). To this end, we propose a corrected simplicial Hausdorff distance for point-cloud-generated simplicial complexes. This is the first systematic extension of the classical Hausdorff metric to the category of simplicial complexes, rigorously satisfying the axioms of an extended metric while being simultaneously sensitive to both geometric structure and topological connectivity. We further construct a computable version of this distance on filtered complexes, analyze its time complexity, and establish the necessity and constraints imposed by monotonicity of the underlying filtration function on the distance definition. Experiments demonstrate that the proposed metric robustly captures structural evolution across filtration scales, offering a new tool for simplicial complex comparison, stability analysis, and machine learning embedding in TDA.

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.

Existing diversity metrics for datasets predominantly rely on statistical distributions or entropy, often overlooking the intrinsic geometric structure of the data. This work introduces persistence landscapes (PLs)—a tool from topological data analysis—into diversity assessment, offering a geometric perspective to quantify structural diversity and establishing a direct link between geometric features and diversity. The proposed PLDiv metric is grounded in rigorous theoretical foundations and exhibits strong interpretability. Empirical evaluations across multimodal settings demonstrate its robustness and reliability, positioning it as a novel paradigm for dataset construction, augmentation, and evaluation.

High-dimensional data dimensionality reduction (DR) often introduces subtle visual artifacts—such as spurious voids or false clusters—in 2D projections, leading to misinterpretation despite apparent structural fidelity. Existing projection quality metrics (PQMs) emphasize structural preservation (e.g., neighborhood or distance retention) but neglect distortions in inter-point spacing distributions. To address this gap, we propose the Warping Index (WI), the first PQM explicitly designed to quantify visual deformation via *inter-point gap preservation*. WI leverages geometric properties of projections by measuring the deviation between local void distributions in the 2D embedding and the corresponding high-dimensional neighborhood structure. Extensive evaluation across diverse DR methods—including t-SNE, UMAP, and PCA—demonstrates that WI significantly outperforms conventional metrics (e.g., Trustworthiness, MR, NL) in detecting misleading voids and structural distortions. It thus provides a reliable, interpretable basis for trustworthy visualization assessment and informed DR method selection.

This work addresses the challenge that geometric and topological anomalies—such as self-intersections and edge collapses—in intermediate wireframes often render generated B-Rep models invalid, while retraining large generative models remains prohibitively expensive. To circumvent this, the authors propose the Wireframe Debugging and Repair (WDR) framework, which intervenes at the intermediate wireframe stage without requiring model retraining. WDR first identifies anomalies by combining a vision-language model for coarse filtering with a Geometry-Topology Anomaly Detector (GTAD), then performs test-time optimization via an Energy-Guided Geometry-Topology Repair (EGGTR) module coupled with diffusion-based resampling. This approach establishes the first training-free pipeline for wireframe anomaly detection and repair, significantly improving B-Rep kernel validation success rates while preserving the diversity and distributional fidelity of generated CAD models.

This work addresses a central challenge in quantum topological data analysis (TDA): how to effectively preserve topological invariants—particularly persistent homology—when encoding classical datasets into quantum states. The authors propose a direct quantum encoding approach that bypasses the conventional construction of simplicial complexes, instead focusing on admissible encoding strategies derived directly from raw distance data. By integrating tools from algebraic topology, metric geometry, and quantum information theory, they systematically analyze how different encoding schemes affect persistent homology structures and identify several quantum encodings capable of efficiently preserving essential topological features. This framework substantially reduces computational resource requirements, offering a new paradigm for low-complexity quantum TDA.