Local Geometry
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Related Articles from SNS
Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry
new Abstract: We present an algorithmic framework for incremental maintenance of first sheaf cohomology $H^1(X; \mathcal{F})$ on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of $H^1$ via factorization of the coboundary matrix requires $O(n^3)$ time; when the complex evolves with a stream of $m$ edits, full recomputation after each edit costs $O(mn^3)$. Under a bounded local geometry assumption -- bounded...
Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry
arXiv:2606.04227v2 Announce Type: replace Abstract: We present an algorithmic framework for incremental maintenance of first sheaf cohomology $H^1(X; \mathcal{F})$ on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of $H^1$ via factorization of the coboundary matrix requires $O(n^3)$ time; when the complex evolves with a stream of $m$ edits, full recomputation after each edit costs $O(mn^3)$. Under a bounded...
From Local Geometry to Global Pseudo Labeling for Robust Positive Unlabeled Learning under Covariate Shift
arXiv:2605.31187v1 Announce Type: new Abstract: Detecting covariate shift is critical for building reliable vision systems. While most prior work focuses on improving robustness to shift, explicitly detecting covariate shift remains underexplored. Existing approaches typically rely on fully supervised training, requiring labeled examples from both original and shifted distributions, which is often impractical.
Local Coordination and the Geometry of Social Networks
Announce Type: replace-cross Abstract: We study agents playing a pure coordination game on a large social network. Agents are restricted to coordinate locally, without access to a global communication device, and so different regions of the network will converge to different actions, precluding perfect coordination. We show that the extent of this inefficiency depends on the network geometry: on some networks, near-perfect efficiency is achievable, while on others welfare is strictly bounded...
Emergent Ordinal Geometry in Transformers Trained on Local Comparisons
arXiv:2606.01269v2 Announce Type: replace Abstract: Transitive inference is the challenge of inferring that A < C from knowing only adjacent relations (A < B, B < C). It is solved by humans and animals not through logical chaining but via an analogue mental number line, whose signature is the symbolic distance effect: distant comparisons are easier than nearby ones. We ask whether Transformers acquire the same primitive, training small models exclusively on adjacent comparisons from a hidden...
Emergent Ordinal Geometry in Transformers Trained on Local Comparisons
arXiv:2606.01269v1 Announce Type: new Abstract: Transitive inference is the challenge of inferring that A < C from knowing only adjacent relations (A < B, B < C). It is solved by humans and animals not through logical chaining but via an analogue mental number line, whose signature is the symbolic distance effect: distant comparisons are easier than nearby ones. We ask whether Transformers acquire the same primitive, training small models exclusively on adjacent comparisons from a hidden...
Skeletal-Anchored Dual Harmonics for Structured 3D Modeling
Announce Type: new Abstract: We present Skeletal-Anchored Dual Harmonics (SADH), a novel 3D shape representation that tightly couples local surface geometry with internal meso-skeletal organization. SADH represents a shape as a collection of compact surface patches rooted on internal anchors optimized directly inside the object volume. Each patch is parameterized using a dual-channel spherical harmonic (SH) formulation, where one channel models local radial geometry while the other defines...
SurGe: Improved Surface Geometry in Point Maps
arXiv:2605.31577v1 Announce Type: new Abstract: Recent feedforward 3D reconstruction methods predict point maps and estimate global 3D geometry remarkably well. However, their predictions still exhibit inaccurate local surface geometry, which is clearly visible qualitatively but only weakly reflected in common metrics. To make these errors more explicit in evaluation, we introduce a point map normal metric that evaluates the local surface orientation induced by neighboring 3D predictions.
HDSL: A Hierarchical Domain-Specific Language for Structured 3D Indoor Scene Generation and Localized Editing with LLM Agents
arXiv:2606.09738v1 Announce Type: new Abstract: Text-driven indoor scene generation and editing require an intermediate representation that language models can both produce and revise. Existing LLM-based systems often rely on scene graphs or global constraint lists, which are compact but underspecify local geometry and make instruction-based edits difficult to localize. We frame this problem as structured program generation and local program repair, and propose Hierarchical Descriptive Scene...
Finding Kissing Numbers with Game-theoretic Reinforcement Learning
arXiv:2511.13391v4 Announce Type: replace Abstract: Since Isaac Newton first studied the Kissing Number Problem in 1694, determining the maximal number of non-overlapping spheres around a central sphere has remained a defining challenge in discrete geometry. As the local analogue of Hilbert's 18th problem, it has profound implications across geometry, number theory and information theory. Although lattices and codes have achieved significant progress, the field is confined to isolated...