The paper proves a compactification for polyhedral norms.
problem Horofunction compactification for polyhedral norms.
method Establishing a criterion for converging sequences and generalizing the moment map.
result The horofunction compactification of a polyhedral norm is homeomorphic to the dual unit ball.
Geometrically connects toric varieties to normed spaces.
problem Connecting algebraic geometry with convex analysis.
method Establishes a 1-1 correspondence using topological models.
result Toric varieties correspond to horofunction compactifications of polyhedral norms.
This work describes compactifications of metric spaces and vector spaces using asymmetric norms.
problem Compactifying metric spaces and vector spaces using asymmetric norms.
method Nonstandard methods, ultrapowers of the spaces at hand.
result Polyhedral compactifications of vector spaces with stratified structure.
Study normal curves in sub-Finsler Lie groups with specific norms, focusing on branching and face stability.
problem Analyzing normal curves in sub-Finsler Lie groups with different norms.
method Using tools from convex analysis, the Pontryagin Maximum Principle is revisited to express the normal equation as a differential inclusion involving the subdifferential of the dual norm.
result Normal curves in polyhedral norms have controls that locally take values in a single face of a sphere with respect to the norm.
Paper tackles image reconstruction from limited data using polyhedral norms and convex regularizers.
problem Learning convex regularizers for image reconstruction from limited data.
method Imposes amplitude-equivariance, approximates functionals with polyhedral norms, identifies synthesis and analysis forms, proposes a trainable tight frame architecture.
result Proposed framework outperforms sparsity-based methods in denoising and biomedical image reconstruction.
We study the stable norm on the first homology of a closed, non-orientable surface equipped with a Riemannian metric. We prove that in every conformal class there exists a metric whose stable norm is polyhedral. Furthermore the stable norm is never strictly convex if the first Betti number of the surface is greater tha…
Method provides bounds for sparse PCA and nuclear norm problems.
problem Semidefinite optimization problems (SDOs).
method Cutting-plane method with focus on initial outer approximation as a second-order cone approximation.
result Method provides bound gaps of 0.5-6.5% for sparse PCA problems with 1000 covariates and solves nuclear norm problems over 500x500 matrices.
The real homology of a compact Riemannian manifold M is naturally endowed with the stable norm. The stable norm on H1(M,R) arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finite-dimensional vector space H1(M,R) are st…
GeoCert finds robustness of neural networks by fitting a ball in polytopes.
problem Computing exact robustness of neural networks for all norms.
method GeoCert algorithm that finds the largest ball within a polytope.
result GeoCert efficiently computes robustness bounds for neural networks.
For a Riemannian polyhedra, we study the geometry of the unit ball for the unidimensional stable norm (stable ball). In the case of a unidimensional Riemannian polyhedra (graph), we show that the stable ball is a polytope whose vertices are completely described by combinatorial properties of the graph. We study then th…
Algorithms compute length spectra of torus graphs efficiently.
problem Computing length spectra of graphs embedded on a torus.
method Preprocessing and algorithms based on polyhedral norms.
result Efficient computation of length spectra and spectrum comparison.
Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turned into a convex optimization problem by replacing the cardinality function by its convex envelope (tightest c…
New theory assesses smoothness of polyhedral surfaces.
problem Evaluate smoothness of polyhedral surfaces.
method Incorporates geometry of polyhedral surfaces, proposes new notions of smoothness.
result Seemingly mild conditions significantly limit polyhedral surface shapes.
Study of hyperbolic polyhedral surfaces with regular faces.
problem Understanding the properties of hyperbolic polyhedral surfaces with regular faces.
method Combinatorial and geometric analysis of hyperbolic polyhedral surfaces with regular faces.
result There is a gap between the areas of non-smooth hyperbolic polyhedral surfaces and smooth hyperbolic surfaces.
Study on discrete Gaussian curvature for polyhedral surfaces.
problem Discretization of Gaussian curvature for polyhedral surfaces.
method Generalization of discrete conformal equivalence to define discrete Gaussian curvature and classify polyhedral surfaces.
result Existence of polyhedral surfaces with constant discrete Gaussian curvature in every discrete conformal class.
A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying …
A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the foll…
SOC-ICNN expands neural network representational capacity by using conic optimization.
problem Restrictive representational capacity of ReLU-based ICNNs.
method Proposes SOC-ICNN architecture that uses Second-Order Cone Programming.
result SOC-ICNN strictly expands representational space without increasing complexity.
Polyhedral semantics for intermediate logics; Nerve Criterion ensures completeness.
problem Characterize polyhedrally-complete intermediate logics.
method Developed Nerve Criterion to characterize polyhedrally-complete logics combinatorially.
result Nerve Criterion provides a necessary and sufficient condition for polyhedrally-completeness.
New algorithm achieves optimal privacy and efficiency in non-Euclidean convex optimization.
problem Optimizing convex functions while maintaining privacy in non-Euclidean settings.
method Developed a linear-time algorithm for ℓp-setups, leveraging geometric properties. result Optimal excess risk achieved in linear time for 1<p≤2. Locally finite complexes with polyhedral CAT(0) metrics are arborescent.
problem Characterizing locally finite complexes with CAT(0) metrics. method Proving arborescence for complexes with polyhedral CAT(0) metrics. result Locally finite complexes with polyhedral CAT(0) metrics are arborescent. The paper proves rigidity of bordered polyhedral surfaces using variational principles.
problem Determining the rigidity of bordered polyhedral surfaces.
method Using the variational principle, the paper shows that bordered polyhedral surfaces are determined by boundary values and discrete curvatures on interior edges.
result The paper re-proves the classical result that two Euclidean or hyperbolic cyclic polygons are congruent if their side lengths are equal.
We develop a method to find a set of diminimal polyhedral maps on the torus from which all other polyhedral maps on the torus may be generated by face splitting and vertex splitting. We employ this method, though not to its completion, to find 53 diminimal polyhedral maps on the Torus.
Characterizes metrics on triangulated surfaces using glued Euclidean triangles.
problem Describing metrics on triangulated surfaces constructed from glued Euclidean triangles.
method Carefully constructing polyhedral metrics and proving their uniqueness.
result Polyhedral metrics are the only intrinsic metrics preserving Euclidean triangle lengths.
Disproves polyhedral immersions of two specific triangulations on a non-orientable surface.
problem Proving non-existence of polyhedral immersions for triangulated surfaces.
method Developed method to disprove existence of polyhedral immersions in R^3.
result Two vertex-minimal, neighborly triangulations of a non-orientable surface are not realizable as polyhedral surfaces in R^3.
A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature me…
A polyhedral map is called {p,q}-equivelar if each face has p edges and each vertex belongs to q faces. In 1983, it was shown that there exist infinitely many geometrically realizable {p,q}-equivelar polyhedral maps if q>p=4, p>q=4 or q−3>p=3. It was shown in 2001 that there exist infi…
Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.
problem Approximating smooth 2-tori in high-dimensional spaces.
method Polyhedral approximation using Lagrangian and isotropic tori.
result Smooth 2-tori can be approximated by polyhedral Lagrangian or isotropic tori in C0 or C1 sense.
Enhances neural network robustness with polyhedral envelope regularization.
problem Improving neural network robustness against adversarial attacks.
method Introduces polyhedral envelope regularization to bound the robustness region.
result Demonstrates improved robustness guarantees with minimal computational overhead.
Survey on discrete curvature concepts for polygons and polyhedral surfaces.
problem Defining curvature for discrete structures like polygons and polyhedral surfaces.
method Explains curvature notions for polygons, polyhedral surfaces, and abstract polyhedral manifolds.
result Discrete curvature theorems parallel classical theorems in differential geometry.
Finite element method approximates scalar curvature in arbitrary dimensions.
problem Approximating scalar curvature using finite elements in arbitrary dimensions.
method Piecewise polynomial interpolants of a smooth Riemannian metric on a triangulated polyhedral domain.
result Finite element interpolants converge to scalar curvature with rate O(hr+1) in H−2(Ω) norm. Proposes a mixture of expert architecture for polyhedral classifiers.
problem Learning polyhedral classifiers with high accuracy.
method Uses an expectation maximization algorithm to learn parameters.
result Generalization bounds are derived and the method performs comparably to state-of-the-art approaches.
The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.
problem Approximating Riemannian manifolds with polyhedral metrics.
method Conditions on curvature tensors for Lipschitz and local polyhedral approximations.
result Conditions are sufficient for local polyhedral approximations, conjectured to be sufficient for global approximations.
Study calculates Floer homology for binary polyhedral spaces.
problem Calculating Floer homology for specific polyhedral spaces.
method Equivariant instanton Floer homology, modified algebraic construction.
result Equivariant instanton Floer homology values for binary polyhedral spaces.
We develop a framework for consistent polyhedral surrogates in classification and prediction.
problem Designing consistent polyhedral surrogates for classification and prediction problems.
method Formalizing and studying embeddings of predictions as points in R^d, assigning original loss values, and convexifying to create surrogates.
result Established a strong connection between embeddings and polyhedral surrogates, providing constructions and proofs of consistency or inconsistency.
We show that area minimizing polyhedral surfaces are saddle.
We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived …
This paper investigates several global rigidity issues for polyhedral surfaces including inversive distance circle packings. Inversive distance circle packings are polyhedral surfaces introduced by P. Bowers and K. Stephenson as a generalization of Andreev-Thurston's circle packing. They conjectured that inversive dist…
New bounds show polyhedral surrogates are optimal for generalization.
problem Proving generalization rates for polyhedral loss functions.
method Developed two general results for polyhedral surrogates.
result Polyhedral surrogates provide linear surrogate regret bounds, translating directly to target rates.
The paper approximates Einstein tensor using finite elements.
problem Approximating Einstein tensor for piecewise polynomial metrics.
method Finite element method applied to Riemannian metrics.
result Convergence rate of O(hr+1) in H−2(Ω)-norm. New index theory proves Gromov's dihedral conjectures.
problem Comparisons and rigidity of scalar curvatures, mean curvatures, and dihedral angles.
method Developed a new index theory for manifolds with polyhedral boundary.
result Proved Gromov's dihedral extremality and rigidity conjectures.
Study of Green function and Laplacians on polyhedral surfaces, focusing on genus two with a conical point.
problem Analyzing the behavior of Green function and self-adjoint Laplacians on polyhedral surfaces.
method Explicit construction of a basis in the kernel of the adjoint Laplacian, computation of S-matrix, study of various self-adjoint extensions.
result The behavior of the S-matrix at the zero value of the spectral parameter is sensitive to the geometry of the polyhedron.
Constructs a moment map flow for isotropic maps on surfaces.
problem Understanding isotropic maps on surfaces and their properties.
method Develops a Kähler moment map geometry and a modified moment map flow.
result Polyhedral modified moment map flow induces a strong deformation retraction.
We investigate the rigidity of hyperbolic cone metrics on 3-manifolds which are isometric gluing of ideal and hyper-ideal tetrahedra in hyperbolic spaces. These metrics will be called ideal and hyper-ideal hyperbolic polyhedral metrics. It is shown that a hyper-ideal hyperbolic polyhedral metric is determined up to i…
Researchers decompose hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.
problem Decomposing hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.
method Two different approaches to demonstrate the existence of polyhedral decompositions.
result The number of polyhedral decompositions of M is finite. In this article we introduce the notion of Polyhedral Kahler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4-dimensional case, prove that such manifolds are smooth complex surfaces, and classify the singularities of the metric. The singularities form a divisor and the res…
The study connects polyhedral manifolds to Riemannian ones with geometric bounds.
problem Connecting polyhedral manifolds to Riemannian manifolds with geometric constraints.
method Using a theorem by C. Lange and B. Bowditch, the study bounds the curvature and injectivity radius of Riemannian manifolds.
result Polyhedral manifolds with bounded geometry are bi-Lipschitz homeomorphic to Riemannian manifolds with controlled curvature and injectivity radius.
Polyhedral theory offers a new approach to designing neural networks.
problem Designing optimal neural network architectures for various applications.
method Polyhedral theory and mixed-integer representability.
result Analytical approach to neural network design.