arXiv research

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

Study proves boundedness of operators in variable exponent Morrey spaces.

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 critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.

Bayesian neural network achieves nearly optimal performance in Besov space.

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

Study on nonlinear elliptic equations with variable exponents, proving existence and multiplicity of solutions.

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.

Systemic risk measures are crucial for the stability of financial markets, yet classical formulations fail to capture the complexity of market volatility. We propose a new framework for systemic risk measurement on the variable-exponent Bochner-Lebesgue space $L^{p(\cdot)}$, where the exponent $p(\cdot)$ is a random va…

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…

DFIV uses deep features for IV regression, achieving optimal rates.

Since the quasiconvex risk measures is a bigger class than the well known convex risk measures, the study of quasiconvex risk measures makes sense especially in the financial markets with volatility. In this paper, we will study the quasiconvex risk measures defined on a special space $L^{p(\cdot)}$ where the variable …

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.

Regularization is used to find a solution that both fits the data and is sufficiently smooth, and thereby is very effective for designing and refining learning algorithms. But the influence of its exponent remains poorly understood. In particular, it is unclear how the exponent of the reproducing kernel Hilbert space~(…

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

We examine random variables in the power law/regularly varying class with stochastic tail exponent, the exponent $α$ having its own distribution. We show the effect of stochasticity of $α$ on the expectation and higher moments of the random variable. For instance, the moments of a right-tailed or right-asymmetric varia…

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

Study of deep neural networks using finite-time Lyapunov exponents.

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

New groups found with critical exponents close to but less than max.

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

We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over $\mathbb{C}$ attached to character polynomials. Our approach interprets the stabilization as a probabilistic phenomenon based on the asymptotic independence of certain *motivic random variables*, an…

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