arXiv research

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.

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

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.

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

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…

Deep learning has exhibited superior performance for various tasks, especially for high-dimensional datasets, such as images. To understand this property, we investigate the approximation and estimation ability of deep learning on anisotropic Besov spaces. The anisotropic Besov space is characterized by direction-depen…

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

Study on stochastic covariant derivatives in curved space-time.

Deep ReLU networks can approximate and learn smooth functions efficiently.

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

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

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

Diffusion models achieve nearly optimal distribution estimation in various spaces.

New spatiotemporal Besov process improves CT image reconstruction and other inverse problems.

The restricted class of quasicircles sometimes called the "Weil-Petersson-class" has been a subject of interest in the last decade. In this paper we establish a Sokhotski-Plemelj jump formula for WP-class quasicircles, for boundary data in a certain conformally invariant Besov space. We show that this Besov space is pr…

We study the expressivity of deep neural networks. Measuring a network's complexity by its number of connections or by its number of neurons, we consider the class of functions for which the error of best approximation with networks of a given complexity decays at a certain rate when increasing the complexity budget. U…

The paper studies harmonic map heat flow stability and decay rates.

ConvResNets approximate Besov functions and classify on low-dimensional manifolds.

Study on distributed nonparametric function estimation with optimal rate and cost of adaptation.

In this paper, we consider various tensorial estimates in geometric Besov-type norms on a one-parameter foliation of surfaces with evolving geometries. Moreover, we wish to do this with only very weak control on these geometries. Several of these estimates were established in previous works by S. Klainerman and I. Rodn…

We prove the local existence of unique smooth solutions of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. It is a semiflow on the Besov space $B^{1,p}_2(M, Λ^2)$ for $p > 4$. The Donaldson geometric flow was introduced by Simon Dona…

The Volterra square-root process shows non-uniqueness of limiting distributions and regularity of its law.

No trapped surfaces can form under low-regularity bounds in certain spacetimes.

Paper proves higher-order flow matching preserves optimality in generative modeling.

Paper studies optimal federated learning for nonparametric regression with privacy constraints.

The paper defines function spaces on manifolds with bounded or singular geometries.

Estimates mixing coefficients of geometrically ergodic Markov processes from a single sample path.

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…

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 …

We show that the complex Monge-Ampere equation on a compact Kaehler manifold (X,ω) of dimension n admits a Holder continuous omega-psh solution if and only if its right-hand side is a positive measure with Holder continuous super-potential. This property is true in particular when the measure has locally Holder continu…

Paper reconciles minimax rates and optimal recovery rates for noisy observations.

We study fractional Sobolev and Besov spaces on noncompact Riemannian manifolds with bounded geometry. Usually, these spaces are defined via geodesic normal coordinates which, depending on the problem at hand, may often not be the best choice. We consider a more general definition subject to different local coordinates…

We study the problem of estimating a nonparametric probability density under a large family of losses called Besov IPMs, which include, for example, $\mathcal{L}^p$ distances, total variation distance, and generalizations of both Wasserstein and Kolmogorov-Smirnov distances. For a wide variety of settings, we provide b…

This work introduces a new probabilistic process for regularization in machine learning.

Deep neural networks approximate functions in shift-invariant spaces with controlled error.

We develop a geometric invariant Littlewood-Paley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited regularity assumptions on the metric. We give invariant descriptions of Sobolev …

Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.

We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ of linear partial differential operators, and we prove the density of the corresponding smooth Sobolev sect…