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Found 22 papers, 3 papers with code
Understanding Scaling Laws with Statistical and Approximation Theory for Transformer Neural Networks on Intrinsically Low-dimensional Data
Our theory predicts a power law between the generalization error and both the training data size and the network size for transformers, where the power depends on the intrinsic dimension $d$ of the training data.
Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study
We establish a theoretical framework to quantify the neural scaling laws by analyzing its approximation and generalization errors.
Deep Neural Networks are Adaptive to Function Regularity and Data Distribution in Approximation and Estimation
This class encompasses a range of function types, such as functions with uniform regularity and discontinuous functions.
Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learning
The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data.
DFU: scale-robust diffusion model for zero-shot super-resolution image generation
Diffusion generative models have achieved remarkable success in generating images with a fixed resolution.
Effective Minkowski Dimension of Deep Nonparametric Regression: Function Approximation and Statistical Theories
Existing theories on deep nonparametric regression have shown that when the input data lie on a low-dimensional manifold, deep neural networks can adapt to the intrinsic data structures.
Deep Nonparametric Estimation of Intrinsic Data Structures by Chart Autoencoders: Generalization Error and Robustness
Our paper establishes statistical guarantees on the generalization error of chart autoencoders, and we demonstrate their denoising capabilities by considering $n$ noisy training samples, along with their noise-free counterparts, on a $d$-dimensional manifold.
On Deep Generative Models for Approximation and Estimation of Distributions on Manifolds
Many existing experiments have demonstrated that generative networks can generate high-dimensional complex data from a low-dimensional easy-to-sample distribution.
High Dimensional Binary Classification under Label Shift: Phase Transition and Regularization
Label Shift has been widely believed to be harmful to the generalization performance of machine learning models.
WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimming
We propose a general and robust framework to recover differential equations using a weak formulation, for both ordinary and partial differential equations (ODEs and PDEs).
Benefits of Overparameterized Convolutional Residual Networks: Function Approximation under Smoothness Constraint
Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness.
A Manifold Two-Sample Test Study: Integral Probability Metric with Neural Networks
Based on the approximation theory of neural networks, we show that the neural network IPM test has the type-II risk in the order of $n^{-(s+\beta)/d}$, which is in the same order of the type-II risk as the H\"older IPM test.
Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces
Learning operators between infinitely dimensional spaces is an important learning task arising in wide applications in machine learning, imaging science, mathematical modeling and simulations, etc.
Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks
Most of existing statistical theories on deep neural networks have sample complexities cursed by the data dimension and therefore cannot well explain the empirical success of deep learning on high-dimensional data.
Multiscale regression on unknown manifolds
We consider the regression problem of estimating functions on $\mathbb{R}^D$ but supported on a $d$-dimensional manifold $ \mathcal{M} \subset \mathbb{R}^D $ with $ d \ll D $.
Doubly Robust Off-Policy Learning on Low-Dimensional Manifolds by Deep Neural Networks
Our theory shows that deep neural networks are adaptive to the low-dimensional geometric structures of the covariates, and partially explains the success of deep learning for causal inference.
Distribution Approximation and Statistical Estimation Guarantees of Generative Adversarial Networks
Generative Adversarial Networks (GANs) have achieved a great success in unsupervised learning.
Learning functions varying along a central subspace
The estimation error of this variance quantity is also given in this paper.
Efficient Approximation of Deep ReLU Networks for Functions on Low Dimensional Manifolds
The network size scales exponentially in the approximation error, with an exponent depending on the intrinsic dimension of the data and the smoothness of the function.
Nonparametric Regression on Low-Dimensional Manifolds using Deep ReLU Networks : Function Approximation and Statistical Recovery
It therefore demonstrates the adaptivity of deep ReLU networks to low-dimensional geometric structures of data, and partially explains the power of deep ReLU networks in tackling high-dimensional data with low-dimensional geometric structures.
IDENT: Identifying Differential Equations with Numerical Time evolution
The new algorithm, called Identifying Differential Equations with Numerical Time evolution (IDENT), is explored for data with non-periodic boundary conditions, noisy data and PDEs with varying coefficients.
Numerical Analysis
Adaptive Geometric Multiscale Approximations for Intrinsically Low-dimensional Data
We consider the problem of efficiently approximating and encoding high-dimensional data sampled from a probability distribution $\rho$ in $\mathbb{R}^D$, that is nearly supported on a $d$-dimensional set $\mathcal{M}$ - for example supported on a $d$-dimensional Riemannian manifold.
