hyperplane arrangement

This study investigates the specification number of Boolean threshold functions—the minimum number of input points required to uniquely determine a function—and its geometric underpinnings. By identifying Boolean threshold functions with chambers in central hyperplane arrangements, the work precisely characterizes the specification number as the number of facets of the corresponding chamber. It establishes deep combinatorial and geometric connections via threshold polytopes and Chow vectors. The main contributions include proving that the average specification number is at most \(2n\), thereby resolving an open question posed by Gutekunst et al. regarding its asymptotic order being \(\Theta(n)\); demonstrating the geometric invariance of minimally specified functions under variable expansion, which answers the fourth operation-related question raised by Lozin et al.; and extending these results to polynomial threshold functions.

This paper addresses the lack of geometric characterization for the implicit spatial partitioning induced by machine learning models. We propose a modeling framework based on Riemannian simplicial complexes: model decision regions are represented as metric-bearing simplicial complexes, enabling systematic quantification of geometric features—including volumes, facet areas, and dihedral angles. To track geometric evolution across neural network layers, we introduce pullbacks of differential forms and an extended Laplacian operator. Furthermore, we define vertex-wise discrete curvature and edge-wise statistical Ricci curvature to explicitly link model geometry with underlying data distributions. The resulting geometric regularization method directly constrains spatial configurations, enhancing both generalization and interpretability. Empirically and theoretically, it demonstrates consistency in regularizer design and efficacy in diagnosing learning dynamics, offering a computationally tractable and principled approach to geometric deep learning.

This paper addresses binary classification of linearly separable data in high-dimensional spaces. Unlike conventional approaches—such as support vector machines (SVMs)—that optimize hyperplane parameters, the proposed method introduces a geometrically driven algorithm centered on a novel “moving-point” mechanism: the separating hyperplane remains fixed, while sample points’ projections in feature space are iteratively adjusted via vector algebra and statistical discriminant criteria to achieve optimal linear separation. The approach integrates vector-space modeling, geometric hyperplane construction, and dynamic point-position optimization, thereby enhancing both geometric interpretability and structural flexibility. Experiments across multiple benchmark datasets demonstrate classification accuracy comparable to SVM, yet with markedly improved transparency and intuitive geometric insight. This work establishes a new geometric paradigm for explainable machine learning.

This study addresses the problem of determining the minimum number of hyperplanes, denoted $S(n)$, required to slice all edges of an $n$-dimensional hypercube. By integrating reasoning large language models with CPro1—a tool featuring automated hyperparameter tuning—we design an efficient search algorithm that yields novel constructive solutions, including an 8-hyperplane slicing scheme for the 10-dimensional hypercube ($Q_{10}$). Our main contributions are a significant improvement of the upper bound on $S(n)$ from $\lceil 5n/6 \rceil$ to $\lceil 4n/5 \rceil$, with a refined bound of $4n/5 + 1$ when $n$ is an odd multiple of 5, and the first non-trivial lower bound on the number of edges that can be sliced by fewer than $n$ hyperplanes. These results substantially advance the theoretical understanding of this classical problem in combinatorial geometry.

This study addresses the efficient enumeration of hole-free row-convex polyominoes. By interpreting the area of a polyomino as an integer partition of its row lengths and employing generating functions to encode horizontal alignments between adjacent rows, the work establishes—for the first time—a direct connection between integer partitions and the enumeration of row-convex polyominoes. This approach yields an exact generating function and leads to the asymptotic formula \( S(N) \approx A \cdot 2^N \cdot \cos(N\theta + \phi) \), where \( \theta = \arctan(\sqrt{7}/3) \). Numerical validation confirms high accuracy for small areas, offering a concise and effective framework for both exact and asymptotic analysis of this class of combinatorial structures.

This work studies the efficient PAC learnability of polytopes defined as the intersection of $k$ halfspaces under the $\rho$-margin assumption. Focusing on continuous distributions where most points lie far from the boundary, the authors propose a novel algorithm whose runtime is $\mathrm{poly}(k, \varepsilon^{-1}, \rho^{-1}) \cdot \exp\left(O(\sqrt{n \log(1/\rho) \log k})\right)$. This is the first result to reduce the exponential dependence on $k$ and $\rho^{-1}$ to a subexponential factor. The method significantly improves upon prior approaches and nearly matches known lower bounds—up to logarithmic factors—in both cryptographic and statistical query models. Furthermore, it broadens the applicability of margin-based learning techniques to a wider class of polytopes.

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 study addresses the construction of functions in algebraic combinatorics subject to stringent distributional constraints and the discovery of previously unknown combinatorial symmetries. To this end, we propose the SLURP framework, which integrates MapSeek-Functional and MapSeek-Symbolic approaches through alternating pseudo-label supervised learning, symbolic regression, and formal verification in Lean 4. The framework yields the first combinatorial interpretation of $q,t$-Narayana polynomials based on non-crossing partitions and provides a combinatorial proof of symmetry in previously unresolved cases by leveraging newly discovered statistics. All code and formalized results are publicly released to ensure reproducibility and rigorous verification.

This work addresses the challenge of high-dimensional spherical range counting, which is notoriously hindered by the curse of dimensionality, making it difficult to simultaneously achieve accuracy and efficiency. For weighted point sets, we present the first data structure supporting approximate spherical range counting with an arbitrary stretch factor $1+\varepsilon$ ($\varepsilon>0$), maintaining sublinear query time even when the number of points in the ambiguous region, $t_q$, is sublinear. By integrating techniques from approximate range searching, high-dimensional geometric indexing, and query-driven preprocessing, our approach achieves near-linear space complexity $O(n^{1+o(1)})$ and query time $n^{1-\Theta(\varepsilon^4/\log(1/\varepsilon))} + t_q^{\varrho} n^{1-\varrho}$, where $\varrho = \Theta(\varepsilon^2)$. This significantly outperforms existing methods in both theoretical guarantees and practical scalability.