Generalizes Sobolev IPM for graph-based measures using Orlicz geometric structure.
problem Limitation of Le et al. (2025) framework to Lp geometry. method Generalizes Sobolev IPM through Orlicz geometric structure, employing convex functions to capture nuanced geometric relationships.
result GSI-M reduces to a simple univariate optimization problem, achieving remarkable computational efficiency.
Proves inequality linking function deviation to gradient norm on compact manifolds.
problem Analyzing coupled elliptic systems on compact manifolds.
method Develops a new Poincaré-Sobolev inequality with a density-free reference average.
result Poincaré constant depends on the density's gradient norm.
Learning rates for least-squares regression are typically expressed in terms of L2-norms. In this paper we extend these rates to norms stronger than the L2-norm without requiring the regression function to be contained in the hypothesis space. In the special case of Sobolev reproducing kernel Hilbert spaces used …
Novel algorithm speeds up computation of Sobolev IPM for graph-based probability measures.
problem Efficient computation of Sobolev IPM for graph-based probability measures.
method Established relation between Sobolev norm and weighted Lp-norm, proposed novel regularization, leveraged graph structure. result Proposed regularized Sobolev IPM provides closed-form expression for fast computation.
Sharp inequality on Siegel domain involving weighted norms and sub-Laplacian.
problem Establishing a Sobolev trace inequality on a specific domain.
method Using weighted norms and fractional powers of sub-Laplacian on Heisenberg group.
result Sharp Sobolev trace inequality on Siegel domain involving weighted norms.
Study improves Poincaré-Sobolev inequalities for differential forms.
problem Improving Sobolev space embeddings for differential forms.
method Utilizes Lq,p-cohomology and bi-Lipschitz images to estimate embedding norms. result Estimates for embedding norms in Euclidean balls and their images.
We refine and generalize several interpolation inequalities bounding the Lp norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich distance to μ on a smooth weighted Riemannian manifold satisfying CD(0,∞) condition.
Gradient descent achieves optimal learning for elliptic PDEs via Sobolev norms.
problem Learning elliptic PDEs from noisy data.
method Gradient descent on Sobolev norm objective functions.
result Gradient descent achieves statistical optimality for elliptic PDEs.
Kernel interpolation is inconsistent for norms with smoothness above a constant.
problem Inconsistency of kernel interpolation in reproducing kernel Hilbert spaces.
method Lower bounds for generalization error in Sobolev norms.
result Kernel interpolation is always inconsistent for norms with smoothness above a constant.
Sharp stability of isometries on Heisenberg group proven.
problem Quantitative stability of isometries on the Heisenberg group.
method Proving quasi-isometries close to isometries with specific closeness orders.
result Quasi-isometries of Heisenberg group are close to isometries with specific closeness orders.
Paper shows deep neural networks can approximate Korobov functions nearly optimally.
problem Approximating Korobov functions with deep neural networks.
method Used deep neural networks and measured approximation rates with Lp and H1 norms. result Achieved a super-convergence rate, outperforming traditional methods.
Study on extremizers for Sobolev inequality on curved manifolds.
problem Existence of extremizers for the sharp p-Sobolev inequality on Riemannian manifolds with nonnegative curvature. method Nonsmooth concentration compactness methods and Mosco-convergence results for Cheeger energy.
result Almost extremal functions are close to radial Euclidean bubbles and almost zero globally under nonnegative curvature.
Study extends neural network approximation to time-varying PDEs using Fourier-Lebesgue spaces.
problem Limitation to static PDEs and different time-domain regularity.
method Extend spectral Barron spaces to anisotropic weighted Fourier-Lebesgue spaces, measure approximation error in Bochner-Sobolev norm.
result Established bound on approximation rate for functions in anisotropic weighted Fourier-Lebesgue spaces.
The study shows how to regularize weakly harmonic maps using Sobolev norms and Coulomb frames.
problem Regularity of weakly harmonic maps between Riemannian manifolds.
method New structure equations and Coulomb-frame methods combined with Hardy-BMO duality.
result Sufficient conditions on Sobolev norms ensure full regularity of weakly harmonic maps.
This paper analyzes shallow ReLU networks in L^p and Sobolev spaces, focusing on approximation and generalization.
problem Approximation and generalization of shallow ReLU networks in L^p and Sobolev spaces.
method Spherical harmonic analysis and embeddings into spectral Barron spaces for L^p spaces, path-norm control for Sobolev spaces.
result Minimax-optimal rates for nonparametric regression with shallow ReLU networks under path-norm control.
New bound relaxes uniform gradient norm assumptions for PAC-Bayesian bounds.
problem Generalization bounds with strict assumptions like uniformly bounded loss.
method Relax uniform bounds assumptions to on-average bounded loss and gradient norm.
result Proposes a new generalization bound with a surrogate of model complexity.
We prove the developability and C1,1/2 regularity of W2,2 isometric immersions of n-dimensional domains into Rn+1. As a conclusion we show that any such Sobolev isometry can be approximated by smooth isometries in the W2,2 strong norm, provided the domain is C1 and convex. Both results fail to …
New tensorization theorem for Sobolev spaces on product spaces.
problem Characterize Sobolev spaces on product metric measure spaces.
method Showed two descriptions of Sobolev space on product spaces coincide.
result Norm equivalence and density results for Sobolev spaces.
In this paper, we propose a method for estimating the Sobolev type embedding constant on a domain with minimally smooth boundary. We estimate the embedding constant by constructing an extension operator and computing its operator norm. We also present some examples of estimating the embedding constant for certain domai…
New activation functions achieve arbitrary-accuracy Sobolev approximation by fixed-size neural networks.
problem Approximation of Sobolev functions by neural networks
method Elementary Universal Activation Function and Differentiable Universal Activation Functions
result Arbitrary-accuracy Sobolev approximation by fixed-size neural networks
New method stabilizes machine learning for physics-informed inverse problems.
problem Reconstructing physical quantities from PDE-compliant measurements.
method Physics-informed learning with smooth inductive bias.
result PDE operators stabilize variance and prevent overfitting in fixed dimensions.
Quantitative Sobolev extensions lead to Neumann heat kernel bounds.
problem Bounding Neumann heat kernels for domains with integral Ricci curvature.
method Quantitative Sobolev extension operators and Neumann heat kernel estimates.
result Uniform bounds on Neumann heat kernels and eigenvalues.
The paper provides approximation guarantees for neural networks trained with gradient flow.
problem Approximating neural networks trained with gradient flow in continuous L2(Sd−1)-norm. method NTK argument for non-convex second but last layer, under-parametrized regime.
result Gradient flow convergence guarantees for neural networks under Sobolev smoothness assumptions.
Error estimates for nonlinear PDEs using kernel/GP methods.
problem Error analysis of kernel/GP methods for nonlinear and parametric PDEs.
method Sobolev space error estimates based on minimizing norm property of the solution.
result Dimension-benign convergence rates for smooth solutions.
We show in this note that the Sobolev Discrepancy introduced in Mroueh et al in the context of generative adversarial networks, is actually the weighted negative Sobolev norm ∣∣.∣∣H˙−1(νq), that is known to linearize the Wasserstein W2 distance and plays a fundamental role in the dynamic formulation of…
Scheme minimizes p-elastic energy of curves over time.
problem Minimizing p-elastic energy of curves over time. method Minimizing movement scheme with approximate normal graphs.
result Short-time existence and lower bound on solution's lifetime.
Optimal rates for vector-valued regression on various norms.
problem Optimal rates for vector-valued ridge regression on continuous norms.
method Combining standard capacity assumptions with tensor product constructions of vector-valued interpolation spaces.
result Optimal rates for vector-valued ridge regression, independent of output space dimension.
The use of certain critical-exponent Sobolev norms is an important feature of methods employed by Taubes to solve the anti-self-dual and similar non-linear elliptic partial differential equations. Indeed, the estimates one can obtain using these critical-exponent norms appear to be the best possible when one needs to b…
Unified framework for Sobolev spaces on vector bundles, including explicit integration by parts.
problem Developing a comprehensive theory for Sobolev spaces on vector bundles.
method Explicit higher-order geometric integration by parts formula on arbitrary Riemannian manifolds.
result Direct proofs of classical theorems in Sobolev spaces on vector bundles.
We prove lifting theorems for complex representations V of finite groups G. Let σ=(σ1,…,σn) be a minimal system of homogeneous basic invariants and let d be their maximal degree. We prove that any continuous map f:Rm→V such that f=σ∘f is of class $C^{…
Constructs non-asymptotic confidence regions for unknown functions in RKHS.
problem Global probabilistic confidence regions for unknown functions in RKHS.
method Reduces confidence region construction to estimating RKHS norm.
result Valid confidence regions can be constructed non-asymptotically.
Let (X,P) be a toric variety. In this note, we show that the C0-norm of the Calabi flow φ(t) on X is uniformly bounded in [0,T) if the Sobolev constant of φ(t) is uniformly bounded in [0,T). We also show that if (X,P) is uniform K-stable, then the modified Calabi flow converges expone…
The paper proves rigidity and vanishing theorems for translating solitons.
problem Understanding the properties of translating solitons in geometry.
method Using Sobolev inequalities and Lq-norms, the paper proves rigidity and vanishing theorems. result Translating solitons are shown to be hypersurfaces under certain conditions.
Let (M,g) be a noncompact complete n-manifold with harmonic curvature and positive Sobolev constant. Assume that L2 norms of Weyl curvature and traceless Ricci curvature are finite. We prove that (M,g) is Einstein if n≥5 and Ln/2 norms of Weyl curvature and traceless Ricci curvature are small enough…
The paper extends inequalities for convex bodies to higher dimensions and various norms.
problem Extending inequalities for convex bodies to higher dimensions and various norms.
method Developed new operators and inequalities for higher-order Lp norms. result Established mth-order Lp isoperimetric inequalities. Study shows harmful overfitting in Sobolev spaces even as training data grows.
problem Harmful overfitting in Sobolev spaces under noisy conditions.
method Geometric argument using Sobolev inequalities.
result Approximately norm-minimizing interpolators exhibit harmful overfitting.
Optimal multiscale learning of linear operators
problem Statistical and computational limits of learning bounded linear operators between Sobolev spaces
method Reformulate as an infinite-dimensional matrix regression problem with heterogeneous multiscale structure
result Establish minimax rates and construct a finite-resolution blockwise least-squares estimator attaining these rates
Study bounds for Brownian motion on manifolds with sticky boundary conditions.
problem Proving geometric bounds for Brownian motion on manifolds with sticky boundary conditions.
method Interpolation involving energy interactions between boundary and interior of the manifold.
result Explicit geometric bounds on Steklov eigenvalues, boundary trace operators, and boundary trace logarithmic Sobolev constants.
The group Diff(M) of diffeomorphisms of a closed manifold M is naturally equipped with various right-invariant Sobolev norms Ws,p. Recent work showed that for sufficiently weak norms, the geodesic distance collapses completely (namely, when sp≤dimM and s<1). B…
The L 1-Sobolev inequality states that the L n/(n--1)-norm of a compactly supported function on Euclidean n-space is controlled by the L 1-norm of its gradient. The generalization to differential forms (due to Lanzani & Stein and Bourgain & Brezis) is recent, and states that a the L n/(n--1)-norm of a compactly support…
A fundamental theorem of Wolfe isometrically identifies the space of flat differential forms of dimension m in Rn with the space of flat m-cochains, that is, the dual space of flat chains of dimension m in Rn. The main purpose of the present paper is to generalize Wolfe's theorem to the se…
Optimizes sharp curvature inequality on spheres, proving near-minimizers are close to standard metric.
problem Optimizing total σ2-curvature on spheres with positive scalar curvature. method Analyzes metrics conformal to the standard sphere, uses Sobolev norms to measure closeness.
result Near-minimizers of total σ2-curvature are almost the standard metric (up to Möbius transformations). We consider the moduli space of the extremal Kähler metrics on compact manifolds. We show that under the conditions of two-sided total volume bounds, L2n-norm bounds on $\Riem$, and Sobolev constant bounds, this Moduli space can be compactified by including (reduced) orbifolds with finitely many singularities…
The paper extends von Neumann's theory to normed modules and shows how they can be represented.
problem Understanding the structure of normed modules and their representability.
method Combining von Neumann's theory of liftings with Gigli's differential structure.
result Every separable normed module can be represented as sections of a measurable Banach bundle.
The study proves conditions for nontrivial solutions on Riemannian manifolds.
problem Conditions for nontrivial solutions to the Dirac equation on Riemannian manifolds.
method Proves a necessary criterion using the Yamabe invariant and Sobolev constant.
result Sharp conditions on the sphere for nontrivial solutions.
We consider maps into Riemannian manifolds of non-positive curvature and start developing a systematic PDE theory. We control the Sobolev H2,2-norm of such a map in terms of its energy, the L2-norm of its tension field and a topological term depending on the homotopy class. We also solve a Dirchlet problem with…
Stability of singularity formation in Yang-Mills fields in higher dimensions.
problem Stability of self-similar blowup profiles for Yang-Mills equations in (1+d)-dimensions. method Analysis of explicitly known equivariant self-similar blowup solution and small equivariant perturbations.
result Global-in-space asymptotic stability of the self-similar blowup solution for Yang-Mills equations in (1+d)-dimensions for d≥5. We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradi…