equiangular tight frame preprocessing

This paper investigates the binder structure of doubly transitive equiangular tight frames (DTETFs). The central problem is to determine whether the binder is empty or forms a balanced incomplete block design (BIBD). Employing tools from group actions, symplectic and quadratic forms over finite fields, combinatorial design theory, and Naimark complements, we establish—rigorously for the first time—that the binder of any DTETF must either be empty or constitute a BIBD. Furthermore, we show that the binder of its Naimark complement corresponds precisely to an oval in the associated BIBD, thereby forging novel connections between DTETFs, oval geometry, and affine Lagrangian subspaces. Finally, we completely characterize the binder properties of four families of binary quadratic-form-based DTETFs: two families admit nonempty binders only in finitely many exceptional cases, while the other two always yield BIBDs. This work unifies and generalizes existing ETF constructions based on symplectic forms and quadratic forms.

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.

Existing normalization layers in convolutional neural networks lack a rigorous theoretical analysis of continuous translational equivariance. Method: We establish the first formal mathematical framework for equivariance in normalization layers by integrating group action theory with signal sampling analysis. We introduce a dual-equivariance definition—“discrete shift + continuous translation”—derive necessary and sufficient conditions for translational equivariance, and uncover its dimension-dependent mechanism. Contribution/Results: We prove that standard normalization schemes—including BatchNorm, LayerNorm, and InstanceNorm—are inherently non-equivariant due to cross-dimensional statistical aggregation over spatial or channel dimensions. Empirical validation on ResNet-18/ImageNet feature maps fully corroborates our theory, precisely delineating the equivariance boundaries of each normalization across dimensions. This work provides the first verifiable, physics-informed design principle for equivariant normalization layers in CNNs.

This work investigates the generalization capabilities of tabular foundation models in cross-modal settings and introduces a unified evaluation framework accompanied by a standardized classification pipeline. The approach leverages equiangular tight frame (ETF) preprocessing, in-context learning, and probability calibration to systematically assess model performance across 95 datasets spanning seven distinct modalities. A novel validation-free ETF-based training stopping criterion is proposed, along with a lightweight baseline built upon frozen features. The method achieves performance comparable to task-specific fine-tuned models on most benchmarks while accelerating inference by 4–200× and producing well-calibrated confidence estimates, thereby substantially enhancing practical deployability.

To address the challenge of modeling correspondences in panoramic dense matching—exacerbated by inherent distortions in equirectangular projection (ERP)—this paper proposes the first end-to-end learning framework grounded in spherical geometry. Our method replaces conventional planar positional embeddings with a novel spherical positional encoding based on 3D Cartesian coordinates of the unit sphere. It further introduces a bidirectional spherical–Cartesian coordinate transformation mechanism to enable distortion-free feature mapping, and incorporates a geodesic flow optimization module for fine-grained matching refinement directly on the sphere. Evaluated on Matterport3D and Stanford2D3D, our approach achieves state-of-the-art performance, improving AUC@5° by 26.72 and 42.62 points, respectively. These gains demonstrate substantial mitigation of ERP-induced geometric distortion, establishing a new benchmark for panoramic dense matching.

This work addresses the geometric inaccuracies in sparse-view filament-based wireframe 3D printing caused by deformation. We propose a Gaussian alignment framework anchored on parametric curves, which constrains Gaussian kernels to these curves to yield a compact, geometry-aware wireframe representation that substantially reduces reconstruction ambiguity under sparse observations. By integrating differentiable rendering with neural deformation field estimation, our method achieves globally consistent deformation alignment and drives the co-evolution of a digital twin model. This enables dynamic updating of printing paths within a closed-loop adaptive control system, robustly compensating for deformations during robotic wireframe fabrication and significantly enhancing both manufacturing accuracy and adaptability for complex structures.

Although equivariant networks are parameter-efficient, their computational cost rivals that of non-equivariant layers due to the unfolding of structured weights into dense matrices. This work decouples equivariant linear layers into a cyclic convolution over the group dimension and a linear transformation along the channel dimension. Leveraging the Fourier convolution theorem and the conjugate symmetry of the real-valued discrete Fourier transform, we introduce the first efficient frequency-domain acceleration algorithm for such layers. Custom CUDA kernels enable full forward and backward passes in both FP32 and FP16 precision. At the operator level, our method achieves up to 2× speedup over PyTorch’s F.linear, and end-to-end models (Flash EQ-ViT and EQ-Swin) attain up to 1.7× faster inference—marking the first time equivariant networks simultaneously surpass non-equivariant counterparts in accuracy, parameter efficiency, and inference speed.

本文解决了3D数据机器学习中的旋转等变性问题，通过几何深度学习、群论和表示论的方法，介绍了实现旋转等变性的现代架构和技术。

This work addresses the parameter redundancy in conventional Rotary Position Embedding (RoPE) within the query, key, and value projections, which fails to fully exploit the expressive efficiency of complex linear transformations. The authors propose a novel RoPE reformulation grounded in genuine complex linear mappings, seamlessly integrating positional encoding into linear transformations in the complex domain. This approach substantially reduces the number of parameters in the attention module by nearly 50% while preserving model performance almost unchanged, thereby enhancing representational conciseness and interpretability. Empirical evaluations demonstrate that the parameter reduction incurs negligible performance degradation on both in-distribution and out-of-distribution tasks, confirming the method’s superiority in parameter efficiency.

This work addresses the construction of scalable point sets in the high-dimensional unit cube whose two-dimensional coordinate projections all exhibit quasi-uniform distribution. For this purpose, it pioneers a synthesis of algebraic number theory and quasi-Monte Carlo methods, leveraging rational directions on the projective line to unify the parametrization of two novel constructions: Kronecker sequences over cubic number fields and nested rank-1 lattice rules combining real quadratic fields with $p$-adic embeddings. Through techniques from algebraic norm estimation, dual Diophantine approximation, and $p$-adic analysis, the resulting point sets achieve optimal-order lower bounds on separation radii and upper bounds on covering radii for all two-dimensional projections, uniformly across any number of points. Moreover, their mesh ratios remain uniformly bounded, thereby ensuring globally consistent geometric quality and excellent scalability.