arXiv research

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

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

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.

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

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

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

Study minimax rates for binary classifier estimation with margin conditions.

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.

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

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…

Shallow diffusion models learn hidden low-dimensional structures effectively.

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

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

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

Study approximates operator learning for PDEs using Fourier multipliers.

The paper constructs exotic complex projective surfaces using rational blowdowns.

Improved bounds for neural network approximations of functions.

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

New reservoir computing approach handles infinite-dimensional systems.

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…

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

Classifies surfaces with T-singularities and ample canonical class.

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

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

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

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

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

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…

Paper tightens statistical aggregation results using local complexity.

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 …

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.

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…

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

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

This paper analyzes neural network classifiers' performance in binary classification.

Transformers enable in-context learning with guarantees for a wide range of tasks.

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…

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…

This paper extends financial theory to measure learnable market structure under computational constraints.