arXiv research

Deep learning performs well on high-dimensional data with anisotropic smoothness.

Bayesian KANs achieve near-minimax posterior contraction rates in anisotropic Besov spaces.

Deep ReLU networks can approximate and learn smooth functions efficiently.

This paper investigates the nonparametric regression problem using SVMs with anisotropic Gaussian RBF kernels. Under the assumption that the target functions are resided in certain anisotropic Besov spaces, we establish the almost optimal learning rates, more precisely, optimal up to some logarithmic factor, presented …

Bayesian neural networks achieve optimal posterior contraction rates in Besov spaces with intrinsic dimensionality.

A rather complete investigation of anisotropic Bessel potential, Besov, and Hölder spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends' is determined by a singularity function which leads naturally to the study of weigh…

Bayesian nonparametric LABS model adapts to function smoothness in Besov 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…

This paper studies continuum-armed bandits under Besov smoothness conditions and derives minimax rates.

New cyclicity measures defined in weighted Besov spaces, with stability and geometric analysis.

Wavelet-based online learning adapts to noisy Besov spaces with high probability.

The study provides theoretical guarantees for the statistical performance of optimal decision trees.

Study analyzes deep learning's performance on variable exponent Besov space, highlighting adaptivity benefits.

Bayesian neural network achieves nearly optimal performance in Besov space.

Deep ReLU networks approximate functions in Sobolev and Besov spaces efficiently.

Adaptive algorithm minimizes online prediction errors for irregular data.

Deep ReLU networks can efficiently approximate Sobolev and Besov functions.

Characterizes when differential forms have weak exterior derivatives based on limiting behavior of integration over simplices.

Extends Onsager's conjecture to Besov spaces on manifolds with boundary.

In this paper we propose a function space approach to Representation Learning and the analysis of the representation layers in deep learning architectures. We show how to compute a weak-type Besov smoothness index that quantifies the geometry of the clustering in the feature space. This approach was already applied suc…

In this paper, we consider the problem of "hyper-sparse aggregation". Namely, given a dictionary $F = \{f_1, ..., f_M \}$ of functions, we look for an optimal aggregation algorithm that writes $\tilde f = \sum_{j=1}^M θ_j f_j$ with as many zero coefficients $θ_j$ as possible. This problem is of particular interest when…

Paper studies stability of curved surfaces in a half-space.

In this paper, we investigate a holonomy invariant elliptic anisotropic surface energy for hypersurfaces in a complete Riemannian manifold, where "holonomy invariant" means that the elliptic parametric Lagrangian (i.e., a Finsler metric) of the Riemannian manifold used to define the anisotropic surface energy is consta…

Anisotropic minimal graphs over half-spaces are flat.

Kolmogorov-Arnold Networks improve deep learning adaptivity and can approximate Besov functions optimally.

Characterizes paths minimizing anisotropic lengths in Euclidean space.

In this note, we give a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is in analogue with the classical case for isoparametric hypersurfaces in Euclidean spaces. On the other hand, by an example of local…

New Kelvin transform for anisotropic elliptic problems.

Study on stochastic covariant derivatives in curved space-time.

Deep learning models can adaptively estimate functions with varying smoothness using regularization.

Paper extends capillary convex body results to anisotropic setting with Alexandrov-Fenchel inequalities.

Formula proves monotonicity for anisotropic minimal hypersurfaces.

A new metric evaluates generative models by comparing real and generated samples.

Paper extends Wente's result to anisotropic capillary surfaces in half-spaces.

Paper proves flatness of anisotropic minimal graphs in half-spaces.

A general approach to formulation of supergravity in higher order anisotropic superspaces (containing as particular cases different supersymmetric extensions and prolongations of Riemann, Finsler, Lagrange and Kaluza--Klein spaces) is given. We analyze three models of locally anisotropic supergravity.

Motivated by the study of wave fronts in anisotropic media, we propose an incidence geometry of anisotropic spheres in a Finsler-Minkowski space. An anisotropic version of the Laguerre functional is considered. In some circumstances, this functional can be used to determine that two wavefronts observed at distinct time…

Study minimax rates for density estimation under Huber contamination and Besov IPM losses.

Paper analyzes sample complexity for offline RL with deep ReLU networks.

In this paper, we study the anisotropic Minkowski problem. It is a problem of prescribing the anisotropic Gauss-Kronecker curvature for a closed strongly convex hypersurface in Euclidean space as a function on its anisotropic normals in relative or Minkowski geometry. We first formulate such problem to a Monge-Ampére t…

In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure $μ$ that naturall…

Paper proves rigidity results for anisotropic capillary hypersurfaces.

An introduction into the theory of locally anisotropic spaces (modelled as vector bundles provided with compatible nonlinear and distinguished linear connections and metric structures and containing as particular cases different types of Kaluza--Klein and/or extensions of Lagrange and Finsler spaces) is presented. The …

In this paper we consider the evolution of a graph-like hypersurface by anisotropic mean curvature flow, under some restrictions on the anisotropic area integrand. We find interior estimates (in both time and space) on the gradient of such hypersurfaces, depending only on the height of the graph and the anisotropic are…

The study proves that certain minimal surfaces are flat under specific conditions.

Diffusion models achieve nearly optimal distribution estimation in various spaces.

The study finds surfaces with constant anisotropic mean curvature foliated by circles in Euclidean space.

Torsions, curvatures, structure equations and Bianchi identities for locally anisotropic superspaces (containing as particular cases different supersymmetric extensions and prolongations of Riemann, Finsler, Lagrange and Kaluza--Klein spaces) are investigated.