dimension counting

To address the limited geometric expressiveness and poor adaptability of tangent-space metrics in landing algorithms for optimization under orthogonality constraints, this paper introduces a novel family of Riemannian metrics defined on the space of full-rank real matrices, naturally generalizing the β-metric to the Stiefel manifold. Constructed rigorously via differential geometry and matrix manifold theory, the proposed metrics provide a more flexible and geometrically faithful tangent-space metric tailored to the intrinsic structure of the constraint manifold within the landing framework. Compared with conventional choices, the new metrics significantly improve iterative stability and convergence speed. Empirical evaluation on canonical orthogonally constrained problems—including principal component analysis (PCA), the orthogonal Procrustes problem, and sparse subspace learning—demonstrates superior accuracy and faster convergence. This work establishes an extensible geometric modeling paradigm for constrained optimization, advancing the theoretical foundation and practical efficacy of landing-based methods.

In computational geometry, floating-point errors accumulate during intermediate algorithmic steps, compromising the robustness of final results—even when individual geometric predicates are exact. To address this, we propose an indirect predicate framework that pioneers the integration of symbolic–numeric hybrid evaluation into predicate design. Our method employs symbolic preprocessing, interval arithmetic, and adaptive precision control to reformulate geometric decision logic, thereby avoiding direct floating-point evaluation and significantly reducing reliance on costly exact arithmetic. Evaluated on classical constructions—including Delaunay triangulation and Voronoi diagrams—our approach achieves zero robustness failures while delivering 3–5× speedup over exact arithmetic implementations. This work bridges the longstanding trade-off between numerical efficiency and guaranteed correctness, offering both strong robustness and high performance.

This paper systematically extends finite tight frame theory to the quaternionic vector space ℍᵈ, addressing the existence, construction, and classification of equiangular lines—i.e., equiangular and equidimensional subspaces. Methodologically, it integrates quaternionic linear algebra, representation theory of Lie groups, and projection operator theory to establish variational characterizations of quaternionic tight frames, develop group-based frame constructions, and derive projective/Unitary equivalence criteria. It formulates the first quaternionic analogue of Zauner’s conjecture and constructs a reversible migration and symmetric mapping mechanism among equiangular configurations in ℝᵈ, ℂᵈ, and ℍᵈ. Key contributions include: (i) establishing a complete theoretical foundation for quaternionic tight frames; (ii) identifying necessary dimensional constraints for the existence of equiangular lines in ℍᵈ; and (iii) providing a unified analytical and transformational framework for equiangular structures across the three number fields—thereby filling a fundamental theoretical gap in the field.

This paper addresses the problem of deriving upper bounds on the covering number for real algebraic varieties, images of polynomial maps, and semialgebraic sets in Euclidean space. Building upon the classical Yomdin–Comte estimate, we develop a unified, tight upper bound applicable to *nonsmooth* semialgebraic sets—not restricted to smooth varieties—via a significantly simplified proof. Our bound yields exact volume estimates for tubular neighborhoods, bridging a gap left by Lotz’s and Basu–Lerario’s methods, which fail for nonsmooth settings. The resulting theory provides a unifying framework for three key problems: (1) near-optimal covering bounds for low CP-rank tensors; (2) minimal sketching dimension characterization for randomized polynomial optimization; and (3) tight generalization error bounds for deep neural networks with ReLU or rational activation functions—some of which match or improve the best known results in the literature.

Cylindrical Algebraic Decomposition (CAD) is a fundamental tool in real algebraic geometry, yet existing algorithms suffer from poor practical efficiency. Method: This paper reformulates CAD construction from a geometric perspective, modeling it intrinsically as fiber-structure analysis of morphisms between real algebraic varieties. We establish, for the first time, an equivalence between geometric fiber cardinality and the existence of semi-algebraic continuous sections. Leveraging this insight, we systematically exploit equation structure during projection and propose a novel fiber-guided CAD algorithm. Contribution/Results: Our approach yields significant efficiency gains: theoretically, it simplifies decomposition logic by replacing algebraic elimination with geometric fiber analysis; practically, it outperforms state-of-the-art CAD implementations across diverse benchmark problems. The core contribution lies in uncovering the geometric essence of CAD and effecting a paradigm shift—from classical quantifier elimination to geometric fiber-structure reasoning.

This work proposes the notion of “generalized height” in simplices, extending the classical concept—defined as the distance from a vertex to its opposite facet—to arbitrary pairs of opposite faces. The relative position of such face pairs is uniformly characterized via the angle between their affine subspaces. By leveraging generalized cross products and Gram determinants, the authors establish an algebraic relationship between generalized heights and classical heights, and prove that all generalized heights are bounded below by the minimal classical height. This result demonstrates that the standard thickness assumption in simplex quality analysis naturally extends to a broader class of geometric quantities, offering a new theoretical framework for assessing simplex quality and informing triangulations of Riemannian manifolds, thereby supporting the generation of high-quality meshes.

This work proposes the Graphical Algebraic Geometry (GAG) framework, which for the first time rigorously formalizes polynomials, ideals, and affine varieties from commutative algebra using a diagrammatic language. By integrating tools from category theory, (co)span semantics, and algebraic geometry, GAG establishes a universal and complete compositional reasoning system for polynomial constraint satisfaction problems (#CSP). The core contributions include establishing a formal correspondence between #CSP and graph rewriting, uncovering a deep connection between GAG and the qudit ZH quantum graphical calculus, and proving that constraint rewriting in GAG is #P-hard. Furthermore, it is shown that computing amplitudes in the qudit ZH calculus requires only a constant number of oracle queries to GAG, thereby opening a novel pathway for efficient modeling of quantum computations.

This work investigates the generalized covering radius of linear codes from a geometric perspective, introducing for the first time the notion of $(\rho,t)$-saturating sets as their counterpart in finite geometry, thereby unifying classical saturating sets and $t$-strong blocking sets. By integrating tools from finite geometry, the dual Grassmannian criterion, affine methods, and combinatorial configuration techniques, the paper establishes novel connections between coding theory and finite geometry. The main contributions include multiple equivalent characterizations and lower bounds on the size of $(\rho,t)$-saturating sets, along with efficient construction methods derived from strong blocking sets, graphs, and projective configurations. These results provide a systematic framework for the analysis and application of generalized covering radii in coding theory.

This work addresses the long-standing challenge of explicitly constructing near-optimal lossless rank extractors, weak subspace designs, and strong $s$-blocking sets over small finite fields whose size depends only on the rank or codimension. By integrating tools from function field theory, polynomial identity testing, and Fourier analysis based on $\varepsilon$-biased sets, the authors achieve the first explicit near-optimal constructions of these objects over non-prime fields with $q \geq \mathrm{poly}(s)$. Notably, the resulting strong $s$-blocking set has size $O(s(k - s)q^s)$, improving upon the previous exponential bound $2^{O(s^2 \log s)} q^s k$ and matching the non-explicit optimal asymptotics. The paper also presents the first explicit near-optimal constructions for both lossless rank extractors and weak subspace designs in this setting.

This work investigates whether the degree-4 sum-of-squares (SoS) hierarchy, formulated in vector-coordinate encoding, can certify the classical upper bound on the number of mutually unbiased bases (MUBs), namely \(m \leq d+1\). By constructing a degree-4 pseudo-expectation—derived from the expectation over Haar-random orthogonal bases—that satisfies orthogonality and mutual unbiasedness constraints, and verifying its compliance with localization constraints and Hermitian positive semidefinite cross-coherence inequalities, the authors demonstrate for the first time that the degree-4 SoS relaxation in vector-coordinate form cannot rule out the existence of more than \(d+1\) MUBs. Notably, in \(\mathbb{C}^6\), this approach fails to refute the existence of seven MUBs, whereas the same-degree SoS relaxation using projection-coordinate Gram matrix encoding recovers the classical bound, revealing a fundamental disparity in the expressive power of these two encodings under SoS relaxations.