We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in H3, particularly those related to assumptions on boundedness or smallness of a certain weighted version of the Willmore energy. We prove, in particular, that small energy controls C1 boundary regularity. We examin…
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48 results for “C1 regularity”
Paper proves smoothness of solutions to a complex geometric problem.
problem Smoothness of solutions to the degenerate Dual Minkowski problem.
method Inspired by Guan and Li's approach for the Aleksandrov problem, the authors derive estimates.
result Proves solutions are regular.
New L0 norm added to TDA for market analysis.
problem Improving TDA tools for market prediction.
method Defined and applied L0 norm in TDA for four markets.
result Enhanced TDA tools for market analysis.
Differentiable sorting and rank normalization are incompatible, with specific conditions for admissibility.
problem Incompatibility between differentiable sorting and rank normalization.
method Formalized admissibility through monotone invariance, batch independence, and rank-space stability conditions.
result Different gap-sensitive and batchwise relaxations of rank normalization violate the conditions for admissibility.
The paper proves regularity for minimal surfaces near polyhedral cones.
problem Understanding the regularity of minimal surfaces near polyhedral cones.
method Adapting Simon's method and establishing -regularity for minimal varifolds.
result Proves -regularity for minimal varifolds near polyhedral cones.
Stable saddle solutions found for specific dimensions of the Allen-Cahn equation.
problem Stability of saddle solutions for the Allen-Cahn equation in specific dimensions.
method Analyzing the Simons cone and energy functional to confirm saddle solutions' stability.
result Stable saddle solutions found for .
Study confirms regularity for convex functionals with bounded degeneracy set.
problem Confirming regularity for minimizers of convex functionals with small degeneracy set.
method Building on previous work, confirms regularity when is positive and bounded away from finitely many points. Constructs a counterexample in where is strictly convex but degenerates on a Simons cone intersection.
result Confirms regularity for minimizers of convex functionals with small degeneracy set.
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.
The Local Isometric Embedding of Two-Metrics of Low Differentiability in Euclidean Three Spacemath.DG
Proves local isometric embedding of low-differentiability metrics in 3D space.
problem Isometric embedding of metrics of low differentiability in Euclidean 3-space.
method Simplified notation, geodesic and level parameters, solutions of initial value problems for first order non-linear PDEs, classical linear algebraic systems.
result Local isometric embedding exists for metrics of C1 differentiability.
Analytic sub-Riemannian geodesics in 3D are always C1 and analytic except finitely many points.
problem Regularity of sub-Riemannian minimizing geodesics in 3D analytic manifolds.
method Investigation of totally nonholonomic analytic distributions, proof of Hausdorff dimension, and regularity of minimizing geodesics.
result Minimizing sub-Riemannian geodesics in 3D analytic manifolds are C1 and analytic except finitely many points.
We define disentanglement in generative models and prove it's related to identifiable factors.
problem Understanding disentanglement in generative models like VAEs and GANs.
method Characterized disentanglement in smooth generative pushforward models using the SVD of the Jacobian.
result Disentanglement is identifiable under certain conditions on the generator, promoting separable factors.
Any cyclic quadrilateral can fit inside any smooth curve.
problem Inscribing cyclic quadrilaterals in smooth curves.
method Proving geometric properties of cyclic quadrilaterals and smooth curves.
result Cyclic quadrilaterals can be inscribed in any closed convex -curve.
Theorem proves integrability for piecewise-smooth distributions.
problem Integrability of piecewise-smooth distributions.
method Generalizations of Frobenius integrability theorem.
result Sufficient criteria for complete integrability with bi-Lipschitz coordinates.
The division map of principal bundles with groupoid structure and generalized gauge transformationsmath.DG
Motivated by the computations done in \cite{C1}, where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category of principal -bundles over a given manifold , I develop in this paper the same ideas for the…
Revises Schwarzschild manifold rigidity proof for spin manifolds.
problem Rigidity of Schwarzschild manifold for spin manifolds.
method Spinorial proof approach.
result Generalizes and includes classical and recent black hole uniqueness theorems.
Generalizes Hecke algebra for double torus, linking to skein algebra.
problem Understanding algebraic structures on double torus.
method Introducing Heegaard dual operators and Dehn twists.
result Established relationship between Hecke algebra and skein algebra.
Study optimal inequality for contact CR-warped product submanifolds in cosymplectic space forms.
problem Optimal inequality for contact CR-warped product submanifolds in cosymplectic space forms.
method Using Gauss and Codazzi equations, prove an optimal inequality.
result Prove an optimal inequality for contact CR-warped product submanifolds.
Let (X,L) be a polarized Kähler manifold that admits an extremal Kähler metric in c1(L). We show that on a nearby polarized deformation that preserves the symmetry induced by the extremal vector field of (X,L), the modified K-energy is bounded from below. This generalizes a result of Chen, Székelyhidi and Tosatti to ex…
A remarkable and elementary fact that a locally compact set F of Euclidean space is a smooth manifold if and only if the lower and upper paratangent cones to F coincide at every point, is proved. The celebrated von Neumann's result (1929) that a locally compact subgroup of the general linear group is a smooth manifold,…
The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…
Proves existence and uniqueness of solutions to a quaternionic Monge-Ampère equation.
problem Solving the quaternionic Monge-Ampère equation for -quaternionic plurisubharmonic functions on a hyperKähler manifold.
method Proves existence and uniqueness of solutions using a Cherrier-type inequality and and estimates.
result Obtains smooth solutions to the quaternionic Monge-Ampère equation.
New geometric system from Hessian operators offers solutions to geometric problems.
problem Solving geometric problems using Hessian operators.
method Introducing a new differential-geometric system based on -Hessian operators.
result Deduced an a priori -estimate for solutions to the Dirichlet problem for -Hessian equations.
Study on extending curves in sub-Riemannian manifolds with compatibility conditions.
problem Validating Whitney extension property for horizontal curves in sub-Riemannian manifolds.
method Analyzing equiregular and singular sub-Riemannian manifolds, using nilpotent approximation and Lusin-like approximation.
result Extension property holds for horizontal curves in sub-Riemannian manifolds with specific conditions.
We present the recent advances along with an error analysis of the IBM speaker recognition system for conversational speech. Some of the key advancements that contribute to our system include: a nearest-neighbor discriminant analysis (NDA) approach (as opposed to LDA) for intersession variability compensation in the i-…
Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.
problem Analyzing weak immersions with bounded second fundamental forms in a critical Sobolev space.
method Develops analysis of Lipschitz immersions with bounded second fundamental forms in space.
result Proves existence of differential structure from weak immersions with bounded second fundamental forms.
The paper studies fundamental groups of compact Kahler varieties with nef anti-canonical bundles.
problem Understanding the fundamental groups of compact Kahler varieties with specific properties.
method Application of geometric analytic theory of Kahler spaces and study of Albanese maps.
result The fundamental group of compact Kahler varieties with nef anti-canonical bundle is almost Abelian.
AIR-Net adapts low-rank regularization dynamically for better image completion.
problem Fixed low-rank regularization limits adaptability to different images.
method AIR-Net uses adaptive and implicit regularization parameterized by a dynamic Laplacian matrix.
result AIR-Net enhances implicit regularization and outperforms fixed methods in non-uniform missing data scenarios.
A 6-regular triangulation for hyperbolic plane created.
problem Creating a 6-regular triangulation for hyperbolic plane.
method Constructed a 6-regular geodesic triangulation.
result A 6-regular geodesic triangulation of the hyperbolic plane was successfully created.
The article explores toric spaces of regular polyhedra, highlighting rational and non-rational cases.
problem Exploring toric spaces associated with regular convex polyhedra.
method Symplectic and complex toric spaces associated with five regular convex polyhedra.
result The regular dodecahedron and icosahedron cannot be treated via standard toric geometry.
Regularized deep networks improve generalization and robustness.
problem Improving generalization and robustness of deep neural networks.
method Input gradient regularization combined with Lipschitz and adversarial robustness.
result Regularized models show improved adversarial robustness and generalization.
Gradient descent implicitly regularizes neural networks by penalizing large loss gradients.
problem How to optimize deep neural networks without explicit regularization.
method Backward error analysis to calculate implicit gradient regularization and demonstrate its effectiveness empirically.
result Implicit gradient regularization biases gradient descent toward flat minima, improving model robustness and test errors.
Choquet regularization improves exploration in RL.
problem Improving exploration in reinforcement learning.
method Introducing Choquet regularizers to measure and manage exploration, reformulating RL problems and deriving explicit solutions.
result Explicit optimal distributions and Choquet regularizers for various exploratory samplers.
Gradient-coherent strong regularization improves deep neural networks' generalization.
problem Deep neural networks overfit with strong L1/L2 regularization.
method Imposes regularization only when gradients are coherent, using stochastic gradient descent.
result Significantly improves accuracy and compression (up to 9.9x).
The paper explores optimal regularizers for data sources, linking them to star bodies.
problem Understanding optimal regularizers for data sources.
method Investigates optimal regularizers for data distributions using star bodies and dual Brunn-Minkowski theory.
result Identifies optimal regularizers and assesses amenability to convex regularization.
A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In…
The paper proves existence and multiplicity of affine connections on regular manifolds.
problem Existence and multiplicity of affine connections on regular manifolds.
method Regularity theory and properties of the structural presheaf.
result The space of regular affine connections is an affine space of the space of regular -valued 1-forms.
Paper develops a new theory on Wasserstein DRO's variation regularization effect.
problem Developing a new theory for Wasserstein DRO's regularization effect.
method General theory on variation regularization effect of Wasserstein DRO.
result New generalization guarantees for adversarial robust learning.
The paper studies convergence rates of Tsallis entropic regularization in optimal transport.
problem Optimal transport with regularization.
method Γ-convergence and quantization/shadow arguments.
result Derives convergence rate of Tsallis entropic regularization.
PL Morse theory proves strong regularity in low dimensions.
problem Understanding regular and critical points in PL manifolds.
method Introducing homologically and strongly regular points, presenting criteria, and constructing examples.
result In low dimensions , homologically regular points are always strongly regular.
Study uses property elicitation to understand how fairness regularizers affect optimal decisions.
problem Understanding how fairness regularizers change the optimal decision in predictive algorithms.
method Property elicitation to analyze the relationship between loss, regularization, and optimal decision.
result Necessary and sufficient condition for when a property changes with the addition of a regularizer.
Fiedler regularization uses spectral graph theory to improve neural network performance.
problem Improving neural network performance by penalizing weights based on connectivity.
method Uses the Fiedler value of the neural network's graph as a regularization tool, providing theoretical and computational methods.
result Demonstrates Fiedler regularization's effectiveness in improving neural network performance.
New input gradient regularization improves adversarial robustness efficiently.
problem Improving adversarial robustness in machine learning models.
method Derive robustness bounds, implement scaleable input gradient regularization, avoid double backpropagation.
result Input gradient regularization is competitive with adversarial training and avoids gradient obfuscation.
Study on convergence rates for optimal transport with regularization.
problem Convergence analysis of divergence-regularized optimal transport.
method Novel methodology using quantization and martingale couplings.
result Sharp rates for various divergences and transport costs.
Dropout is a simple but effective technique for learning in neural networks and other settings. A sound theoretical understanding of dropout is needed to determine when dropout should be applied and how to use it most effectively. In this paper we continue the exploration of dropout as a regularizer pioneered by Wager,…
We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on Riemannian manifolds. As an application, we show that solutions to the Yamabe flow instantaneously regularize and become real analytic in space and time. The regularity result is obtained by i…
Improved optimal regularity for harmonic almost complex structures.
problem Establishing optimal regularity for harmonic almost complex structures.
method Quantitative stratification method and rectifiability of singular strata.
result Optimal regularity theory for energy minimizing harmonic almost complex structures.
Fast Learning Rate of Non-Sparse Multiple Kernel Learning and Optimal Regularization Strategiesstat.ML
In this paper, we give a new generalization error bound of Multiple Kernel Learning (MKL) for a general class of regularizations, and discuss what kind of regularization gives a favorable predictive accuracy. Our main target in this paper is dense type regularizations including \ellp-MKL. According to the recent numeri…
Study on the regularity of -Gauss curvature flow near flat interfaces.
problem Regularity of -Gauss curvature flow near flat interfaces.
method Analysis of convex hypersurface near the interface.
result Regularity of the convex hypersurface near the interface.