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.
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.
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.
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.
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 …
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…
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.
Optimizes learning Hilbert-Schmidt operators between Sobolev spaces.
problem Statistical limits of learning mappings between infinite-dimensional function spaces.
method Minimax optimal regularization and multilevel training.
result Multilevel kernel operator learning achieves optimal learning rate.
This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the Lp norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds …
Unified kernels for diverse applications in math and stats.
problem Unified kernels for diverse applications in math and stats.
method Unified parametric class of kernels, characterized by Sobolev spaces.
result Unified kernels encompass various known kernels and their properties.
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.
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.
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.
Paper tackles functional linear regression using spectral algorithms with discrete observations.
problem Functional linear regression problem with discretely observed data.
method Combines distributed spectral algorithms with Sobolev kernels for regularization.
result Derives matching upper and lower bounds for convergence in Sobolev norm.
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.
Noiseless KRR achieves optimal rates and exhibits saturation effects.
problem Understanding optimal rates and saturation phenomena in noiseless kernel ridge regression.
method Comprehensive study of noiseless KRR, establishing minimax optimal rates and uncovering phenomena of extra-smoothness and saturation.
result Noiseless KRR achieves minimax optimal rates and exhibits saturation effects.
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.
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.
Characterizes functions representable by infinite-width ReLU networks with bounded weights.
problem Understanding function representation in overparameterized neural networks.
method Analyzes functions in Ws,1(R) spaces and their Radon transform. result All functions in Ws,1(R) can be represented with bounded norm. 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.
A new method bridges explicit and implicit deep generative models using Stein discrepancy.
problem Limitations of explicit and implicit deep generative models.
method Joint training framework that combines an explicit density estimator and an implicit sample generator via Stein discrepancy.
result The method improves the accuracy of density estimation and quality of generated samples.
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.
New method estimates densities using Sobolev regularization, outperforming existing algorithms.
problem Non-parametric density estimation with clear inductive bias.
method Regularizes Sobolev norm of density, approximates kernel via sampling, uses natural gradients for optimization.
result Method ranks second best on ADBench anomaly detection benchmark.
The paper shows how multi-task learning in neural networks is similar to kernel regression and Hilbert spaces.
problem Understanding the solutions to multi-task shallow ReLU neural network learning problems.
method Analyzing the properties of solutions to multi-task shallow ReLU neural network learning problems, proving uniqueness and equivalence to minimum-norm interpolation problems in Hilbert spaces.
result The solutions to multi-task neural network interpolation problems are almost always unique and coincide with the solution to a minimum-norm interpolation problem in a Sobolev (Reproducing Kernel) Hilbert Space.
The study characterizes kernel spaces on hyperspheres, impacting cubature algorithms.
problem Characterizing kernel spaces on hyperspheres for cubature algorithms.
method Characterization of Sobolev spaces and reproducing kernel Hilbert spaces over hyperspheres.
result Direct consequences for kernel cubature and worst-case error rates.
Novel method learns memory kernels in Langevin equations.
problem Estimating memory kernels in Langevin equations.
method Regularized Prony method for correlation functions, followed by regression over Sobolev norm-based loss function with RKHS regularization.
result Method outperforms other regression estimators in exponentially weighted L^2 space.
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.
The paper analyzes kernel classifiers' performance in Sobolev spaces and proves their optimality.
problem Theoretical analysis of kernel classifiers' performance in Sobolev spaces.
method Deriving upper and lower bounds on classification excess risk using kernel regression theory and estimating interpolation smoothness.
result The proposed kernel classifier is optimal in Sobolev spaces, with theoretical bounds confirmed by real data.
Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.
problem Properties of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
method Construction and study of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
result Hypoelliptic logarithmic Sobolev inequalities on the space.
The paper studies the diameter of diffeomorphism groups with Sobolev metrics.
problem Determine the diameter of diffeomorphism groups with right-invariant Sobolev metrics.
method Analyzes various right-invariant Sobolev norms and their effects on the geodesic distance.
result The diameter of the diffeomorphism group is infinite for strong enough norms and finite for weak enough norms.
Study on online nonparametric regression using Sobolev kernel methods.
problem Adversarial nonparametric regression in high dimensions.
method Online kernelized ridge regression with Sobolev kernel analysis.
result Upper bounds on regret for Sobolev space classes, revealing optimality in certain cases.
Sharp fractional Sobolev inequalities on closed manifolds identified.
problem Critical fractional Sobolev embedding on closed Riemannian manifolds.
method Intrinsic heat-kernel based framework, determining optimal coefficients, proving sharp inequalities.
result Sharp p-power inequality and almost sharp inequality established. We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with…
Estimates latent norms and Gram matrices for graphs on Euclidean balls.
problem Estimating latent points and their relationships in graphs on Euclidean balls.
method Estimates latent norms and Gram matrices using observed graph data.
result Graphs on Euclidean balls can have power-law degree distributions.
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.
In this paper, we prove the equivalent of ultracontractive bound of heat semigroup or the uniform upper bound of the heat kernel with the Nash inequality, Log-Sobolev inequalities on graphs. We also show that under the assumption of volume growth and nonnegative curvature CDE′(n,0) the Sobolev inequality, Nash inequa…
Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.
problem Developing an ABP approach to Sobolev and Michael-Simon inequalities under volume noncollapsing assumptions.
method Using a refinement of Brendle's contact-set argument to derive lower bounds for the volumes of geodesic balls.
result A Michael-Simon type inequality for immersed submanifolds with nonnegative sectional curvature and volume noncollapsing.
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.
The paper proves various inequalities on gradient shrinking Ricci solitons.
problem Understanding geometric inequalities on gradient shrinking Ricci solitons.
method Proving multiple inequalities equivalent on complete gradient shrinking Ricci solitons.
result Various inequalities (Sobolev, logarithmic Sobolev, Schrödinger, etc.) are equivalent on gradient shrinking Ricci solitons.
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.
This study reveals fundamental trade-offs between memorization and robustness in neural networks.
problem Understanding the balance between memorization and robustness in neural networks.
method Analyzes two-layer neural networks in various high-dimensional linearized regimes, focusing on Sobolev-seminorm.
result Establishes fundamental trade-offs between memorization and robustness, with lower bounds on Sobolev-seminorm.
Scalable kernel methods for large datasets using Fourier representations and NUFFT.
problem Cubic complexity in kernel methods limits their use on large-scale datasets.
method Fourier representation of kernels combined with NUFFT for O(n log n) complexity.
result Achieves minimax convergence rates and processes up to tens of billions of samples.
New estimator for estimating derivatives in nonparametric regression.
problem Estimating derivatives of regression functions.
method Plug-in kernel ridge regression (KRR) estimator.
result Plug-in property for derivatives estimation, optimal rate of convergence.
In this paper, we first derive a Sobolev inequality along the harmonic-Ricci flow. We then prove a linear parabolic estimate based on the Sobolev inequality and Moser's iteration. As an application, we will obtain an upper bound estimate for the heat kernel under the flow.
New analysis shows a gap between Gaussian RKHS and neural networks on unbounded domains.
problem Understanding the function space bias of neural networks compared to Gaussian RKHS.
method Infinite-center asymptotic analysis of neural network Banach space and Gaussian RKHS on unbounded domains.
result Certain functions in Gaussian RKHS have infinite norm in neural network Banach space on unbounded domains.
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.