arXiv research

This paper introduces a new Barron space for graph signals and proves its properties for GCNNs.

Study embeddings between Barron spaces with various activation functions, focusing on RePU.

New findings suggest Barron space doesn't defy curse of dimensionality for certain types of smoothness.

Paper analyzes DRM for solving high-dimensional elliptic PDEs with generalization bounds.

Group-invariant neural networks improve approximation accuracy for symmetric functions.

The paper establishes a continuous embedding between two types of Barron spaces in neural networks.

Two-layer neural networks can approximate functions with fractal singularities.

Shallow diffusion models learn hidden low-dimensional structures effectively.

Paper finds sparse representation of functions using inverse scale space flow.

Neural networks approximate and estimate binary classifiers with polynomial input dependence.

The paper studies how to recover smooth Barron functions from $L^p$ samples efficiently.

The paper analyzes neural networks for solving high-dimensional Schrödinger eigenvalue problems.

Deep neural networks achieve optimal learning rates for high-dimensional classification.

Neural networks can approximate high-dimensional classifiers with ReLU networks under margin conditions.

One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation errors by a particular machine learning model. In this paper, we address this iss…

Improved bounds for neural network approximations of functions.

Study extends neural network approximation to time-varying PDEs using Fourier-Lebesgue spaces.

Study variation spaces for neural networks, linking them to approximation theory.

This note explains when neural networks can be seen as Gaussian processes.

We analyze random feature and two-layer neural networks using duality framework.

The paper constructs exotic complex projective surfaces using rational blowdowns.

This paper studies neural networks with bounded norms to avoid the curse of dimensionality.

Study minimax rates for binary classifier estimation with margin conditions.

New neural network rates for unbounded domains with weighted Sobolev spaces.

New algorithm reduces regret in online portfolio and quantum state learning.

Study approximates operator learning for PDEs using Fourier multipliers.

We algebraically prove K-stability of polarized Calabi-Yau varieties and canonically polarized varieties with mild singularities. In particular, the} "stable varieties" introduced by Kollar-Shepherd-Barron and Alexeev, which form compact moduli space, are proven to be K-stable although it is well known that they are \t…

Bayesian neural SDEs calibrate financial models robustly.

Study wormholes in surface moduli space, proving conjecture.

The minimum description length (MDL) principle in supervised learning is studied. One of the most important theories for the MDL principle is Barron and Cover's theory (BC theory), which gives a mathematical justification of the MDL principle. The original BC theory, however, can be applied to supervised learning only …

Study improves density estimation for compact domains using $h$-lifted KL divergence.

Paper tightens statistical aggregation results using local complexity.

We study a new class of codes for lossy compression with the squared-error distortion criterion, designed using the statistical framework of high-dimensional linear regression. Codewords are linear combinations of subsets of columns of a design matrix. Called a Sparse Superposition or Sparse Regression codebook, this s…

Study shows Bergman kernels match averages on quotient spaces, proving non-vanishing of Poincaré series.

Wide neural networks learn features under $μ$P, identifying weights and decomposing support.

Sparse superposition codes were recently introduced by Barron and Joseph for reliable communication over the AWGN channel at rates approaching the channel capacity. The codebook is defined in terms of a Gaussian design matrix, and codewords are sparse linear combinations of columns of the matrix. In this paper, we prop…

We construct a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=2$ which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=1$. In order to construct the example, we combin…

Classifies surfaces with T-singularities and ample canonical class.

New reservoir computing approach handles infinite-dimensional systems.

We establish $L^{\infty}$ and $L^2$ error bounds for functions of many variables that are approximated by linear combinations of ReLU (rectified linear unit) and squared ReLU ridge functions with $\ell^1$ and $\ell^0$ controls on their inner and outer parameters. With the squared ReLU ridge function, we show th…

Convolutional neural networks (CNNs) have been shown to achieve optimal approximation and estimation error rates (in minimax sense) in several function classes. However, previous analyzed optimal CNNs are unrealistically wide and difficult to obtain via optimization due to sparse constraints in important function class…

An infinitely wide model is a weighted integration $\int \varphi(x,v) d μ(v)$ of feature maps. This model excels at handling an infinite number of features, and thus it has been adopted to the theoretical study of deep learning. Kernel quadrature is a kernel-based numerical integration scheme developed for fast approxi…

Tree tensor networks balance model complexity and empirical risk for high-dimensional function approximation.

For projective varieties with definite first Chern class we have one type of canonical metric which is called Kähler-Einstein metric. But for varieties with an intermidiate Kodaira dimension we can have several different types of canonical metrics. In this paper we introduce a new notion of canonical metric for varieti…

Affine spiking neural networks learn efficiently and generalize well.

DNNs can learn complex functions efficiently by breaking the curse of dimensionality.

New analysis shows a gap between Gaussian RKHS and neural networks on unbounded domains.

Study exact minimax rates for density estimation over convex classes, extending previous work.