measure decision boundaries

Quantization induces uncontrolled deformations in neural network decision boundaries, compromising model robustness and generalization. This work presents the first systematic characterization of geometric alterations in classifier decision boundaries under finite numerical precision and introduces a boundary-aware quantization strategy. By leveraging geometric and statistical metrics—including local logit-margin radius, boundary displacement, normal vector variation, and sliced Jaccard distance—the method dynamically determines optimal quantization stopping points on a calibration set, integrating PTQ-W with margin-aware rounding optimization. Experiments on CIFAR-10 demonstrate that the approach reduces the prediction flip rate of 6-bit PTQ-W to 5.3% (boundary Jaccard = 0.184); with boundary-aware stopping, the flip rate further drops to 0.0083 at 8 bits (Jaccard = 0.048). Notably, calibration-set boundary metrics exhibit strong correlation with test performance (r = 0.947–0.994).

This paper addresses the challenge of vast and structurally complex decision boundaries in DNA read mapping to reference genomes in next-generation sequencing (NGS). It investigates the boundary properties of naïve Bayes classifiers under graph-structured input spaces. To this end, the authors propose “neighborhood similarity” — a novel uncertainty measure that is both theoretically interpretable and universally computable, overcoming the reliance of conventional Bayesian confidence on model outputs. Leveraging graph-model-driven boundary analysis, neighborhood distribution statistics, and uncertainty quantification, the study reveals the high-dimensional complexity of decision boundaries and proves that the proposed measure simultaneously captures intrinsic Bayesian uncertainty. Moreover, it seamlessly extends to black-box classifiers lacking built-in confidence mechanisms. Empirically, neighborhood similarity significantly enhances classification interpretability and robustness, offering a principled framework for uncertainty-aware read mapping in NGS applications.

This work proposes a computable geometric metric—local surface volume of decision boundaries—derived from differential geometry to quantify the geometric structure of decision boundaries in deep neural networks, thereby explaining their accuracy and generalization capabilities. For the first time, the Weyl tube formula is adapted to high-dimensional deep learning settings and validated on both convolutional and fully connected networks in image classification tasks. Experimental results demonstrate that, in convolutional networks, smaller boundary volumes—indicating smoother decision boundaries—are significantly correlated with higher classification accuracy, whereas fully connected networks exhibit stronger task-dependent behavior. This study establishes a geometric link between model complexity and performance, offering a novel perspective for understanding generalization in deep learning.

Identifying boundary points of compact manifolds (with boundaries) embedded in high-dimensional data remains challenging, particularly under unsupervised settings. Method: This paper proposes an unsupervised, parameter-interpretable boundary detection algorithm. It innovatively integrates the local reconstruction principle of Locally Linear Embedding (LLE) with spectral analysis of local covariance matrices, leveraging eigenvalue distribution to characterize geometric boundaries. A dual-neighborhood scheme—combining ε-balls and K-nearest neighbors—is introduced to adaptively model structural differences between interior and boundary points on the local manifold. Theoretical analysis assigns clear geometric interpretations to key parameters. Results: Extensive evaluation on diverse synthetic manifolds with boundaries demonstrates that the algorithm significantly improves both boundary point recall and precision, outperforming state-of-the-art methods.

Existing boundary point detection methods are sensitive to density heterogeneity and struggle to identify boundary points in concave structures and high-dimensional manifolds, thereby limiting downstream clustering and classification performance. To address this, we propose LoDD (Local Directional Dispersion), a density-agnostic boundary criterion. LoDD innovatively quantifies local directional dispersion by analyzing eigenvalues of the KNN neighborhood’s covariance matrix to characterize directional uniformity. It further introduces a gridded distribution assumption to enable adaptive estimation of critical parameters. The method integrates density-agnostic KNN search, directional feature modeling, and structure-driven parameter optimization. Extensive experiments on synthetic and real-world datasets, deep learning training set partitioning, and point cloud void detection demonstrate that LoDD significantly outperforms state-of-the-art approaches. Code and benchmark datasets are publicly available.

Accurate boundary detection in high-dimensional data remains a central challenge in unsupervised learning, particularly in the presence of non-linear structures and heterogeneous densities. In this work, we introduce Mean Curvature Boundary Points (MCBP), a novel geometric framework grounded in Geometric Machine Learning that departs from traditional density-based approaches by explicitly modeling the intrinsic curvature of the data manifold. The method relies on a discrete approximation of the shape operator, estimated from local k-nearest neighbor patches, to compute pointwise mean curvature without requiring explicit manifold parametrization. The key insight of MCBP is to use mean curvature as a principled descriptor of boundary structure: high-curvature regions naturally correspond to transitions between clusters, geometric irregularities, and low-density interfaces. This yields a unified geometric interpretation of boundary, outlier, and transition points. We further introduce an adaptive percentile-based thresholding scheme that enables multiscale boundary extraction without relying on ad hoc density parameters. Beyond detection, we propose a curvature-driven data decomposition that separates samples into smooth (low-curvature) and boundary (high-curvature) subsets, effectively acting as a non-linear geometric filtering mechanism. This representation enhances cluster separability and improves the robustness of downstream unsupervised algorithms. Extensive experiments on synthetic and real-world datasets demonstrate that MCBP consistently improves clustering performance, particularly in complex and high-dimensional scenarios. These results position MCBP as a concrete contribution to Geometric Machine Learning, highlighting the potential of curvature-aware analysis as a unifying paradigm bridging differential geometry and data-driven modeling.

This work proposes local Urysohn width as an intrinsic measure of the topological-geometric complexity of classification problems, characterizing the minimal number of diameter-constrained local experts required to guarantee correct classification. By integrating tools from algebraic topology (e.g., Betti numbers), metric geometry, and statistical learning theory, the paper establishes a rigorous hierarchy theorem and a topological-geometric scaling law, and demonstrates that this measure is bidirectionally separated from the VC dimension. The central contribution lies in revealing the fundamental constraint imposed by Urysohn width on classifier complexity and deriving a sample complexity lower bound of Ω(w log w) that is independent of the VC dimension.

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.

This study investigates how the geometric structure of decision boundaries in deep classifiers evolves with layer depth and the dynamical mechanisms governing this process. By modeling feedforward networks as non-autonomous discrete dynamical systems, we employ finite-time maximum Lyapunov exponents (FTMLE) to analyze data trajectories, revealing the dynamical characteristics of decision boundaries across probability, logit, and hidden layers. This work establishes the first mathematical connection between the Lyapunov spectrum and decision boundaries, proving that probability-level FTMLE encodes normal geometric information of the boundary. Building on this insight, we propose a geometry-aware fine-tuning strategy that reconstructs sensitivity distributions within hidden layers, thereby providing a theoretical foundation for layer-aware regularization.

This work addresses classification in high-dimensional Hilbert metric spaces by proposing the first support vector machine (SVM) algorithm with polynomial time complexity. By integrating linear programming optimization with geometric modeling based on the Hilbert and Funk metrics, the method efficiently solves both hard-margin and soft-margin SVM problems as well as nearest-neighbor classification tasks. In contrast to prior approaches that either lack theoretical runtime guarantees or exhibit only exponential complexity, this approach achieves polynomial time complexity in the number of samples, ambient dimension, and the number of facets of the underlying convex body. This advancement substantially enhances the scalability and practicality of classification in high-dimensional non-Euclidean spaces.