define decision boundaries

The decision boundaries of classification models are often difficult to visualize, hindering model interpretability. Method: This paper proposes the first dedicated visualization framework that jointly leverages supervised and unsupervised learning—integrating manifold learning, class-aware distance metrics, and boundary-sensitive embedding optimization—to geometrically map discriminative decision boundaries in low-dimensional space, thereby preserving discriminative structure that conventional dimensionality reduction methods tend to obscure. Contribution/Results: Empirical evaluation on chemical neurotoxicity data demonstrates that the generated visualizations clearly reveal complex nonlinear decision boundaries, and their geometric configurations align closely with domain-knowledge-based toxicity mechanisms. This enables both qualitative attribution and quantitative boundary analysis. The framework significantly enhances model diagnostic capability and establishes a generalizable, geometry-driven paradigm for trustworthy AI.

本文通过引入程序路径和转换系统来分析BSS RAMs在初等结构上的决策过程，并提供算法枚举有限的程序路径。

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.

This work addresses the problem of data sufficiency for linear optimization under cost vector uncertainty: identifying the minimal dataset that uniquely determines the optimal decision. Methodologically, it introduces the first geometric sufficiency criterion for linear programming, grounded in convex geometry and duality theory, to characterize the critical cost directions governing optimality; it further establishes a modeling framework for uncertainty sets and designs a task-driven data selection algorithm. Theoretically, it proves the existence of a small-scale, structured minimal cost dataset sufficient to fully recover the optimal solution. This work provides rigorous theoretical guarantees and an efficient constructive procedure for task-aware data acquisition, overcoming key limitations of conventional sufficiency analyses—namely, their reliance on statistical assumptions or large-sample requirements.

This paper addresses the challenge of simultaneously achieving scalability, local interpretability, and multi-attribute, multi-class fairness in rule-based classification models. To this end, we propose the first column generation–based rule learning framework. Methodologically, we introduce column generation—previously unexplored in rule learning—integrating a linear programming master problem, a decision-tree–inspired column generation heuristic, a surrogate pricing subproblem solver, and weighted rule optimization; we further formulate generalized fairness constraints supporting multiple sensitive attributes and multi-class outcomes. Our key contributions are: (1) enabling local interpretability via rule weights, and (2) unifying support for complex fairness constraints and scalable search over large rule spaces. Extensive experiments on benchmark datasets demonstrate that our approach achieves significant trade-off improvements among accuracy, interpretability, and fairness, substantially enhancing the practicality of rule models in real-world, large-scale applications.

This work addresses the limitations of traditional decision boundary maps (DBMs) in high-dimensional settings, where reliance on dimensionality reduction in the original feature space often leads to class overlap and ambiguous visualizations. To overcome this, the study introduces Shapley values into DBM construction for the first time, mapping data into a Shapley value space before applying dimensionality reduction techniques such as t-SNE or UMAP. This approach substantially enhances class separability and structural compactness in the resulting visualizations. Empirical evaluations demonstrate that the proposed method matches or exceeds state-of-the-art alternatives across established visualization quality metrics, yielding decision boundary maps that are not only clearer but also more interpretable—thereby facilitating deeper exploration and understanding of decision-making behaviors in high-dimensional models.

This work investigates the fundamental gap between axis-aligned decision trees and shallow neural networks in terms of geometric parsimony and interpretability, focusing on the inherent tension in accurately approximating decision boundaries. By analyzing infinitely wide, norm-bounded single-hidden-layer ReLU networks through the Radon total variation (RTV) seminorm, the study distinguishes between classification recovery and score learning objectives, revealing for the first time that common smooth surrogates such as the sigmoid still exhibit infinite RTV in high dimensions. The authors innovatively construct a smooth barrier score function with finite RTV that enables exact threshold-based classification. They establish an L¹(P) calibration bound under a tubular mass condition, proving that the calibration error decays polynomially with the sharpness parameter. Experiments validate the trade-offs among model complexity, accuracy, and threshold selection.

This work challenges the conventional view that overlooks the pivotal role of single-layer threshold logic in high-dimensional spaces. It proposes a novel paradigm—integrating threshold units, dimensionality, and depth—by replacing deep architectures with high-dimensional single-layer threshold units, reframing neural computation as navigation within high-dimensional geometry. Drawing upon Cover’s theorem, linear programming, high-dimensional geometry, and Peircean semiotics—particularly the notion of indexicality—the study develops an interdisciplinary model of perceptron behavior. The analysis reveals that in sufficiently high dimensions, a single hyperplane almost always suffices to separate data, and that the essence of deep networks lies in iteratively deforming data to conform to favorable high-dimensional geometric structures. This insight offers a unified explanation for the expressive power of generative AI models.

This project addresses the practical challenges of integrating numerical computation with machine learning in mathematical research by proposing a systematic solution. Methodologically, it combines exterior differential flux formulas, structure-guided neural network design, interval arithmetic residual bounding, and statistical evaluation techniques to explicitly distinguish three forms of evidence: numerical consistency, predictive accuracy, and rigorous bounds. The primary contributions comprise three independently readable practical tutorials accompanied by computational notebooks, which effectively support example reproduction and methodological transfer. By providing standardized guidance for engineering implementation, this work facilitates the adaptation of these integrated approaches across related domains.

This work addresses the limitations of traditional generalization analyses, which rely on the often unverifiable assumption of independent and identically distributed (i.i.d.) data and thus struggle to accurately characterize model performance on unseen data. The paper proposes a deterministic generalization analysis framework that dispenses with any prior probabilistic assumptions. By examining the sensitivity of optimization solutions to data perturbations, it decomposes the generalization error into geometric and probabilistic components, achieving their first-ever decoupling. The framework expresses generalization bounds via a variational principle, leveraging deterministic perturbation analysis and optimization sensitivity theory to capture the discrepancy between in-sample and out-of-sample performance. Error terms are evaluated through posterior statistical hypotheses, enabling the recovery of conventional high-probability or expected generalization guarantees—all without requiring distributional assumptions.