Paper proposes new density estimators for high-dimensional data.
problem Prohibitive computational cost and slow convergence rate in high-dimensional density estimation.
method Adaptive hyperbolic cross density estimators in mixed smooth Sobolev spaces.
result Proposed estimators do not suffer curse of dimensionality under Integral Probability Metrics.
Paper introduces a new regularization method for kernel gradient descent learning.
problem Preventing overfitting in kernel gradient descent learning.
method Random smoothing regularization as novel convolution-based smoothing kernels.
result Optimal convergence rates achieved in various function spaces.
Deep ReLU networks can approximate and learn smooth functions efficiently.
problem Efficiently approximating and learning smooth functions using deep ReLU neural networks.
method Extending recent results to anisotropic and mixed smooth function classes, establishing approximation rates.
result Deep ReLU networks achieve minimax optimal rates up to logarithmic factors for various smooth function classes.
Study partial derivatives on non-smooth metric measure structures.
problem Understanding partial derivatives in non-smooth settings.
method Extension of Schwarz's theorem and analysis of Sobolev regularity.
result Complete set of results relating properties of functions in non-smooth spaces.
Deep learning has shown high performances in various types of tasks from visual recognition to natural language processing, which indicates superior flexibility and adaptivity of deep learning. To understand this phenomenon theoretically, we develop a new approximation and estimation error analysis of deep learning wit…
Uniform volume estimate for Kähler metrics in big cohomology classes.
problem Estimating volume for singular Kähler metrics in big cohomology classes.
method Generalized mixed energy estimate for functions in complex Sobolev space to big cohomology classes.
result Uniform non-collapsing volume estimate for local Kähler metrics.
Survey on smooth function and form density in Riemannian Sobolev spaces.
problem Density of smooth functions and forms in Sobolev spaces on Riemannian manifolds.
method Careful examination of weak covariant derivatives and partial derivatives.
result Equivalence of weak covariant derivatives to weak partial derivatives.
Local smoothing of metrics with small curvature, removing Ricci curvature condition.
problem Establishing local smoothing of metrics with curvature concentration.
method Local mollification, removing Ricci curvature condition, Sobolev constants and volume growth.
result Compactness of manifolds with small curvature concentration under Ahlfors regularity and Sobolev constant.
Study shows Sobolev functions on non-compact manifolds can't be approximated by smooth compactly supported functions.
problem Sobolev functions on non-compact manifolds cannot be approximated by smooth compactly supported functions.
method Analysis of Sobolev spaces on non-compact manifolds.
result Proves the failure of the density of smooth compactly supported functions in Sobolev spaces on non-compact manifolds.
The paper analyzes convergence rates of Langevin dynamics and Proximal Sampler using Φ-divergence.
problem Analyzing convergence rates of Langevin dynamics and Proximal Sampler.
method Extending mixing time analyses to Φ-divergence, using strong data processing inequalities. result Convergence of Φ-divergence to 0 exponentially fast along Unadjusted Langevin Algorithm and Proximal Sampler. 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.
Rapid mixing of Langevin dynamics on Riemannian manifolds
problem Mixing time of Langevin dynamics on Riemannian manifolds
method Relation between Langevin processes in domain and image
result Achievable polynomial mixing times
We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as initial data in certain Sobolev spaces. Thus the space of closed plane…
Smooth activations enable optimal error rates in neural networks for Sobolev function classes.
problem Achieving optimal approximation and estimation error rates for neural networks in Sobolev function classes.
method Study of neural networks with smooth activations, proving optimal rates via approximation and statistical properties.
result Constant-depth networks with smooth activations achieve optimal rates of approximation and estimation, demonstrating smoothness adaptivity.
Develop a comprehensive theory for regularized M-estimation in reproducing kernel Hilbert spaces.
problem Regularized M-estimation in reproducing kernel Hilbert spaces
method Existence and measurability of the estimator, sharp rates of convergence
result New rates for tensor product Sobolev spaces
We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping p…
Proves inequality for tensor fields on curved spaces.
problem Generalizing inequality for tensor fields on curved spaces.
method Alexandrov-Bakelman-Pucci (ABP) method
result Proves Michael-Simon-Sobolev inequality for tensor fields.
Study shows smooth holomorphic structures can be approximated from weak connections.
problem Approximating smooth holomorphic structures from weak connections.
method Proves connections with specific properties can be approximated in Sobolev norms.
result Strong approximations of smooth holomorphic structures from weak connections.
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 compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.
problem Compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.
method Introduced a locally convex topology, extended compactness theorem, studied pseudo-differential operators, and applied to microlocal defect measures.
result Extended microlocal defect measures and compensated compactness theorem to Sobolev wave front set spaces.
We consider the operator algebra generated by pseudodifferential operators on a closed smooth surface and shift operator induced by a Morse--Smale diffeomorphism of this surface. Elements in this algebra are considered as operators in the scale of Sobolev spaces and the aim of this paper is to describe how Fredholm pro…
Paper studies minimax optimal regression using Laplacian smoothing over graphs.
problem Minimax optimal regression over Sobolev spaces.
method Laplacian smoothing on neighborhood graphs.
result Upper bounds match minimax optimal rates for first-order Sobolev class.
Develops nonparametric regression for non-smooth functions using fractional Laplacian.
problem Non-smooth regression functions in high dimensions.
method Fractional Laplacian eigenmaps for L2-fractional Sobolev spaces. result Upper bound on estimation error of $n^{- rac{2s}{2s+d}}$.
Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.
problem Characterizing and proving rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.
method Analysis of Riemannian manifolds and metric measure spaces with synthetic lower Ricci curvature bounds, using concentration compactness and Polya-Szego inequalities.
result Closed Riemannian manifolds with optimal Sobolev constant are isometric to the sphere, and almost equality implies close measure Gromov-Hausdorff convergence to a spherical suspension.
Sharp log-Sobolev inequalities proved for CD(0,N) spaces.
problem Proving log-Sobolev inequalities in noncompact metric measure spaces.
method Sharp isoperimetric inequality, symmetrisation, scaling argument, Hamilton-Jacobi inequality, Sobolev regularity.
result Sharp log-Sobolev inequalities established in CD(0,N) spaces. Study on smoothness of solutions to nonlinear equations on Riemannian manifolds.
problem Smoothness of solutions to nonlinear equations with Neumann boundary conditions on Riemannian manifolds.
method Integral refinement of Bochner's identity.
result Semilinear Calderón-Zygmund type results on Sobolev regularity.
Develops Poisson structures on weak Sobolev loop spaces for integrable systems.
problem Analyzing integrable systems on low regularity loop spaces.
method Extending Mokhov's constructions to weak Sobolev spaces, constructing presymplectic and Poisson structures.
result Valid Poisson structures and deformations for weak Sobolev loops, extending Hamiltonian formalisms.
The paper extends Sobolev spaces to Finsler manifolds and shows density of smooth functions.
problem Defining and analyzing Sobolev spaces on Finsler manifolds.
method Defining Sobolev spaces for Finsler structures and proving density of smooth functions.
result Smooth functions can approximate weak solutions and functions in Sobolev spaces on Finsler manifolds.
The paper establishes bounds on the smoothness parameter in Gaussian process interpolation.
problem Estimating the smoothness parameter in Gaussian process models.
method Approximation theory in Sobolev spaces and general theorems on parameter estimation.
result Maximum likelihood estimation recovers the true smoothness for certain classes of functions.
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.
New metrics on curve spaces improve shape analysis.
problem Discretization of curve spaces and metric completeness.
method Sobolev metrics on discrete regular curves, completeness analysis.
result The finite-dimensional Riemannian manifolds are complete.
A Morse complex for Axiom A flows on smooth manifolds.
problem Constructing a finite-dimensional cohomological complex for Axiom A flows.
method Defining anisotropic Sobolev spaces and spectral projectors.
result The cohomology of the constructed complex is isomorphic to De Rham cohomology.
By means of a general gluing and conformal-deformation construction, we prove that any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannan manifold without boundary. Applications to Sobolev spaces, Nash embedding and loca…
We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, i…
In this paper, we introduce a deformation analysis of index theory over non compact manifolds, by use of new functional spaces which are the reduced version of Sobolev spaces. It allows to construct Fredholm theory for elliptic differential operators over non compact spaces which possibly do not have closed range with …
The paper studies elliptic operators on manifolds with boundary.
problem Characterizing boundary conditions for elliptic operators on manifolds.
method Using Calderón projectors and mixed order Sobolev spaces, the paper describes the space of boundary values and characterizes Fredholm and regular realisations.
result Characterization of boundary conditions for elliptic operators leading to Fredholm and regular realisations.
Characterizes nodal volumes of Gaussian fields on manifolds, extending previous work.
problem Understanding the law and regularity of nodal volumes for Gaussian fields on manifolds.
method Gaussian measures, Morse theory, Malliavin-Sobolev spaces, ray absolute continuity.
result Extension and generalization of previous work on stationary fields to arbitrary dimensions.
The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
problem Gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
method Assumes a Sobolev inequality and integral Ricci bounds, proving local gradient estimates and Liouville type results.
result Proves local gradient estimates and Liouville type results on manifolds with lower bounds of Ricci curvature.
We prove the smoothness of the L^2-analytic torsion form on some fiber bundles with non-compact fibers of positive Novikov-Shubin invariant. We do so by generalizing the arguments of Azzali-Goette-Schick to an appropriate Sobolev space, and proving that the Novikov-Shubin invariant remains positive in the Sobolev setti…
Log Sobolev and Michael Simon inequalities for tensor fields on curved manifolds.
problem Establishing inequalities for tensor fields on curved manifolds.
method Applying the ABP method to symmetric tensor fields on manifolds with nonnegative sectional curvature.
result Log Sobolev and Michael Simon inequalities for tensor fields.
Given a compact manifold Nn, an integer k∈N∗ and an exponent 1≤p<∞, we prove that the class C∞(Qm;Nn) of smooth maps on the cube with values into Nn is dense with respect to the strong topology in the Sobolev space Wk,p(Qm;Nn) when the homotopy group $π_…
We show for a certain class of operators A and holomorphic functions f that the functional calculus A↦f(A) is holomorphic. Using this result we are able to prove that fractional Laplacians (1+Δg)p depend real analytically on the metric g in suitable Sobolev topologies. As an application we obtain loc…
Subtle issues arise when extending homotopy invariants to spaces of functions having little regularity, e.g., Sobolev spaces containing discontinuous functions. Sometimes it is not possible to extend the invariant at all, and sometimes, even when the formulas defining the invariants make sense, they may not have expect…
Simplified proof of Morrey's lemma for metric spaces.
problem Existence of area-minimizing surfaces in metric spaces.
method Simpler proof using Korn and Lichtenstein's smooth isothermal coordinates.
result Strengthened application to Sobolev maps in metric spaces.
New rigidity theorem for sharp spectral gap in nonnegatively curved spaces.
problem Rigidity of sharp spectral gap in nonnegatively curved spaces.
method Mixing Sobolev theory and singular 1D-localization.
result Rigidity of λ=diam2π2 in compact RCD(0,N) spaces. We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip(S^n,Y) are not dense in the Sobolev space W^{1,n}(S^n,Y). On the other hand we show that if a metric space Y is Lipschitz (n-1)-connected, then Lipschitz …
We introduce a family of conformal invariants associated to a smooth metric measure space which generalize the relationship between the Yamabe constant and the best constant for the Sobolev inequality to the best constants for Gagliardo-Nirenberg-Sobolev inequalities ∥w∣˚q≤C∥∇w∥2θ∥w∥p1−θ. Thes…
Transformer networks approximate Hölder and Sobolev functions with fixed-depth networks.
problem Nonparametric regression with dependent observations.
method Established novel upper bounds for Transformer networks approximating Hölder and Sobolev functions under various β-mixing data assumptions. result Explicit convergence rates for nonparametric regression problems under β-mixing data assumptions.