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.

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.

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

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

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

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

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…

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

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

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

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

Shallow diffusion models learn hidden low-dimensional structures effectively.

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

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

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

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

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

Study approximates operator learning for PDEs using Fourier multipliers.

Study wormholes in surface moduli space, proving conjecture.

The paper constructs exotic complex projective surfaces using rational blowdowns.

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…

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

Improved bounds for neural network approximations of functions.

Study minimax rates for binary classifier estimation with margin conditions.

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

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

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…

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

Classifies surfaces with T-singularities and ample canonical class.

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 …

This work defines a new function space for multi-layer neural networks.

New reservoir computing approach handles infinite-dimensional systems.

This paper analyzes shallow ReLU networks in L^p and Sobolev spaces, focusing on approximation and generalization.

New model classes for function approximation by neural networks defined on domains.

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

New metrics avoid high-dimensional analysis challenges, proving convergence without 'curse of dimensionality'.

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…

Let $X$ be a canonically polarized variety, i.e. a complex projective variety such that its canonical class $K_{X}$ defines an ample $\Q-$line bundle, and satisfying the conditions $G_1$ and $S_2$. Our main result says that $X$ admits a Kähler-Einstein metric iff $X$ has semi-log canonical singularities i.e. iff $X$ is…

Sharp lower bounds on shallow neural networks' approximation rates are derived.

Bayesian neural SDEs calibrate financial models robustly.

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…

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…

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