New CSIM index improves image patch recovery from missing data.
problem Recovering missing image samples using sparse representation.
method Proposes a new convex similarity index (CSIM) and an iterative sparse recovery method.
result Proves the convergence of the algorithm to the globally optimal solution.
Paper proposes a new method for recovering missing samples in images.
problem Missing sample recovery in image signals.
method Iterative sparse recovery algorithm using constrained l 1 l_1 l 1 -norm minimization with a new CSIM fidelity metric. result Simulation results demonstrate the efficiency of the proposed method.
A method classifies image-sets using convex cones based on CNN features.
problem Image-set classification using CNN features.
method Modeling CNN features as convex cones and measuring geometric similarity.
result Enhanced classification through discriminant space maximization of between-class variance.
The paper characterizes sets with infinite hyperbolic convex hull volume.
problem Characterizing sets with infinite hyperbolic convex hull volume.
method Geometric conditions and self-similar sets.
result Characterizes continua and planar self-similar sets with infinite hyperbolic convex hull volume.
This paper studies quasar-convex functions to improve optimization methods.
problem Improving optimization methods for non-convex functions.
method Study of first order methods for quasar-convex functions.
result Proves complexity upper bounds similar to convex functions.
The study examines nilpotent similarity structures on manifolds and their properties.
problem Characterizing closed manifolds with nilpotent similarity structures.
method Generalizes convexity arguments to geodesic segments in nilpotent Lie groups.
result Closed manifolds with nilpotent similarity structures are either complete or radiant.
We classify the self-similar solutions to a class of Weingarten curvature flow of connected compact convex hypersurfaces, isometrically immersed into space forms with non-positive curvature, and obtain a new characterization of a sphere in a Euclidean space R n + 1 \R^{n+1} R n + 1 .
Extends convex clustering to graph-structured data.
problem Handling graph-structured data with convex clustering.
method Formulates a convex objective and uses a proximal dual algorithm for efficient recovery.
result Demonstrates the effectiveness of the method on real-life datasets.
In this work we define a new pseudometric in K ∗ n \mathcal K^n_* K ∗ n , the hyperspace of all non-degenerated compact convex sets of R n \mathbb R^n R n , which is invariant under similarities. We will prove that the quotient space generated by this pseudometric (which is the orbit space generated by the natural action of the group of…
The paper proves conditions for self-similar solutions of curvature flows to be round spheres.
problem Conditions for strictly convex self-similar solutions of curvature flows to be round spheres.
method Employing a new inequality, the paper shows curvature pinching conditions and compares curvature functions.
result Conditions for self-similar solutions of curvature flows to be round spheres.
Proves stability of convex spheres with similar geodesic lengths.
problem Stability of convex spheres with specific geodesic properties.
method Proves C^0 Cheeger-Gromov closeness to the round sphere.
result Strictly convex 2-spheres are close to the round sphere.
New method replaces traditional convex integration for solving geometric problems.
problem Constructing solutions with self-similarity properties in geometric embeddings.
method Introducing Kuiper differential relations and a Corrugation Process to replace traditional convex integration.
result Totally real isometric embeddings exhibit self-similarity and can be uniformly expressed.
New method clusters neurons with similar connectivity profiles.
problem Accurately determining which neurons have similar neurological tasks.
method Proposes clustered Gaussian graphical model and symmetric convex clustering penalty.
result Demonstrates effectiveness of the approach on synthetic and real-world data.
The study finds that only round spheres shrink self-similarly under certain curvature flows.
problem Investigating self-similar solutions to curvature flows by high powers of curvature.
method Analyzing closed strictly convex hypersurfaces in R n + 1 \mathbb{R}^{n+1} R n + 1 under specific curvature flows. result Only round spheres shrink self-similarly under the studied curvature flows.
We study the contraction of a convex immersed plane curve with speed (1/α)k^{α}, where αin(0,1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic self-similar solution. We also discuss a special symmetric case of type two blow-up and show that it converge…
We show the uniqueness of strictly convex closed smooth self-similar solutions to the α α α -Gauss curvature flow with ( 1 / n ) < α < 1 + ( 1 / n ) (1/n) < α< 1+(1/n) ( 1/ n ) < α < 1 + ( 1/ n ) . We introduce a Pogorelov type computation, and then we apply the strong maximum principle. Our work combined with earlier works on the Gauss Curvature flow imply that the α α α -Gauss c…
Study on volumes of quasifuchsian manifolds, focusing on similarities and proximities.
problem Understanding the relationship between renormalized volume and dual volume of quasifuchsian manifolds.
method Analyzing similarities and proximities between renormalized volume and dual volume, using variational formulas and Weil-Petersson distance.
result Renormalized volume and dual volume are closely related, with bounded distance between related objects.
New meta-learning algorithm improves on classical methods with strong guarantees.
problem Improving gradient-based meta-learning methods for better performance and efficiency.
method Developed a new meta-algorithm for online convex optimization, combining gradient-based and regularization-based approaches.
result The algorithm achieves strong sample efficiency and generalization bounds, matching a theoretical lower bound.
On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces with the Weil-Petersson distance function. A set of similarities the resulting met…
We study the fundamental limits to communication-efficient distributed methods for convex learning and optimization, under different assumptions on the information available to individual machines, and the types of functions considered. We identify cases where existing algorithms are already worst-case optimal, as well…
The abstract investigates convexity in locally conformally symplectic geometry.
problem Characterizing and proving convexity in locally conformally symplectic manifolds.
method Geometric characterization and proof of convexity theorems for twisted and symplectic moment maps.
result Established an analog of the symplectic convexity theorem for locally conformally symplectic manifolds.
The paper defines quasi-convex subsets in spaces with lower curvature bound.
problem Understanding the geometry of spaces with lower curvature bound.
method Introducing and exploring quasi-convex subsets in Alexandrov spaces.
result Quasi-convex subsets are a fundamental concept for comparing Riemannian and Alexandrov spaces.
Adaptive meta-learning improves few-shot learning and federated learning performance.
problem Improving few-shot learning and federated learning performance.
method Adaptive gradient-based meta-learning methods integrating online convex optimization and sequential prediction algorithms.
result Improved meta-test-time performance on standard problems in few-shot learning and federated learning.
Set-functions appear in many areas of computer science and applied mathematics, such as machine learning, computer vision, operations research or electrical networks. Among these set-functions, submodular functions play an important role, similar to convex functions on vector spaces. In this tutorial, the theory of sub…
Pseudo-Anosov subgroups in surface bundles over tori are convex cocompact.
problem Understanding the structure of pseudo-Anosov subgroups in surface bundles over tori.
method Using the Birman exact sequence to show convex cocompactness.
result Finitely generated, purely pseudo-Anosov subgroups are convex cocompact in surface bundles over tori.
The paper examines self-similar solutions in warped products.
problem Analyzing self-similar solutions in warped products.
method Investigates solutions satisfying F − F = g ˉ ( λ ( r ) ∂ r , ν ) F-\mathcal{F}=\bar{g}(λ(r)\partial_{r},ν) F − F = g ˉ ( λ ( r ) ∂ r , ν ) , focusing on slices and uniqueness in specific spaces. result Slices are the only closed strictly convex self-similar solutions in the hemisphere for certain curvature functions.
Two new algorithms improve federated optimization under second-order similarity.
problem Federated learning under communication constraints and second-order similarity.
method SVRP and Catalyzed SVRP algorithms combining proximal point evaluations, client sampling, and variance reduction.
result Achieves superior performance and uniformly improves upon existing algorithms for federated optimization under second-order similarity and strong convexity.
Improved IHT with momentum accelerates convex optimization with non-convex constraints.
problem Optimizing convex criteria with non-convex constraints.
method Modified iterative hard thresholding with momentum.
result Acceleration leads to significant improvements over state-of-the-art methods.
Study finds solutions to flows by negative curvature powers.
problem Curvature flows with negative powers.
method Closed self-similar solutions in warped product manifolds, proving non-strict convexity.
result Proves self-similar solutions are slices of warped product manifolds.
Geodesic tomography identifies piecewise constants on convex manifolds.
problem Determining piecewise constant functions on nontrapping manifolds.
method Iterating local uniqueness results based on geodesic integrals.
result Piecewise constant functions are uniquely determined by their geodesic integrals.
The renormalized volume is reinterpreted using isoperimetric profiles.
problem Understanding the renormalized volume of convex co-compact hyperbolic 3-manifolds.
method Using isoperimetric profiles and Minkowski inequalities.
result A sharp Minkowski inequality for horospherically convex sets in H 3 \mathbb{H}^3 H 3 . SignSVRG improves SignSGD by reducing variance, achieving similar convergence rates.
problem Minimizing finite sums of convex and Lipschitz functions.
method Incorporates variance reduction techniques into SignSGD.
result Achieves convergence rates of O ( 1 / T ) \mathcal{O}(1 / \sqrt{T}) O ( 1/ T ) for expected norm of the gradient and O ( 1 / T ) \mathcal{O}(1/T) O ( 1/ T ) for smooth convex functions. Proves a similar inequality to a conjecture about hyperbolic space hypersurfaces.
problem Proving a conjecture about weighted Alexandrov-Fenchel inequalities for hyperbolic space hypersurfaces.
method Analyzes horospherically convex hypersurfaces in hyperbolic space.
result Proves a similar inequality to the conjectured one, provides a counterexample when applicable.
Study of spacelike discs in Minkowski cones, proving self-similar expansion.
problem Mean curvature flow of spacelike discs in Minkowski cones.
method Analysis of parabolic boundary value problem for self-similar solutions.
result Existence of solutions rescaling to self-similarly expanding solutions.
The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.
problem Conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.
method Proving Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in Euclidean and Minkowski contexts.
result Zero Gaussian curvature convex hypersurfaces must be hyperplanes if the mean curvature goes to zero at infinity.
Paper proposes FedAMP for improved federated learning with non-IID data.
problem Challenges of federated learning with non-IID data.
method FedAMP: Federated attentive message passing for pairwise client collaborations.
result FedAMP improves federated learning performance with non-IID data.
Survey on geometry of co-Minkowski space and its affine deformations.
problem Understanding the geometry of co-Minkowski space and its affine deformations.
method Affine deformations of hyperbolic lattices acting on co-Minkowski space, convex core, mean hypersurface, asymmetric norm.
result Existence of a unique mean hypersurface and an asymmetric norm on the space of affine deformations.
Presented are two neural network architectures for convex functions, demonstrating competitive performance.
problem Approximating convex functions efficiently and accurately.
method Developed two neural network architectures: one based on linear-by-part representation and the other on cubic splines.
result Cubic ICKAN networks produce results similar to classical ICNNs in solving convex approximation problems.
Solves convex extension problems for 1-jets and convex hypersurfaces.
problem Finding convex extensions for 1-jets and hypersurfaces.
method Provides necessary and sufficient conditions for convex extensions with C 1 C^1 C 1 smoothness. result Necessary and sufficient conditions for convex extensions of 1-jets and hypersurfaces.
We prove that convex hypersurfaces in R n + 1 {\mathbb R}^{n+1} R n + 1 contracting under the flow by any power α > 1 n + 2 α>\frac{1}{n+2} α > n + 2 1 of the Gauss curvature converge (after rescaling to fixed volume) to a limit which is a smooth, uniformly convex self-similar contracting solution of the flow. Under additional central symmetry of the ini…
We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes span…
Study cash-subadditive risk measures without quasi-convexity.
problem Cash subadditivity without quasi-convexity.
method Represent cash-subadditive risk measures as lower envelopes of quasi-convex measures and introduce quasi-star-shapedness.
result General cash-subadditive risk measures can be represented as lower envelopes of quasi-convex measures.
We give a simple proof that the Frank-Wolfe algorithm obtains a stationary point at a rate of O ( 1 / t ) O(1/\sqrt{t}) O ( 1/ t ) on non-convex objectives with a Lipschitz continuous gradient. Our analysis is affine invariant and is the first, to the best of our knowledge, giving a similar rate to what was already proven for projected gra…
Study on non-Gromov hyperbolic tube domains and their geometric properties.
problem Characterizing non-Gromov hyperbolic tube domains with convex bases.
method Provided a criterion for non-Gromov hyperbolicity, studied Hilbert metric, and continuity properties of complex geodesics.
result Similarity of geometry of tube domains and convex domains, connections between metrics.
Study conjugate locus in convex 3-manifolds using Jacobi fields.
problem Understanding conjugate points in convex 3-manifolds.
method Use Jacobi fields to define coordinate systems and classify singularities.
result Developed a novel method for determining conjugate points in 3D manifolds.
Classifies self-similar solutions for heat equations with positive speed.
problem Classifying self-similar solutions for semilinear heat equations.
method Analyzes the semilinear heat equation u t = Δ u + ∣ u ∣ p − 1 u u_t=Δu+|u|^{p-1}u u t = Δ u + ∣ u ∣ p − 1 u for p > 1 p>1 p > 1 . result Finite time blowing up solutions converge to a positive constant after rescaling.
Harmonic functions on compact symmetric spaces exhibit strong convexity properties.
problem Understanding the convexity of harmonic functions on compact symmetric spaces.
method Analyzing the nonnegativity of the Laplacian powers of harmonic functions.
result Harmonic functions on compact symmetric spaces have nonnegative Laplacian powers, demonstrating strong convexity.
Convex neural networks enforce convex constraints on weights and activations, improving generalization.
problem Improving generalization and reducing overfitting in neural networks.
method Enforce convex constraints on weights and activations, using non-negative weights and non-decreasing convex activation functions.
result Convex neural networks self-regularize, outperforming base architectures and achieving similar performance to convolutional architectures.