On the Representation of Solutions to Elliptic PDEs in Barron Spaces

Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of $d$-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the integral of certain parametric ridge function against a probability measure on the parameters. We prove under some appropriate assumptions that if the coefficients and the source term of the elliptic PDE lie in Barron spaces, then the solution of the PDE is $\epsilon$-close with respect to the $H^1$ norm to a Barron function. Moreover, we prove dimension-explicit bounds for the Barron norm of this approximate solution, depending at most polynomially on the dimension $d$ of the PDE. As a direct consequence of the complexity estimates, the solution of the PDE can be approximated on any bounded domain by a two-layer neural network with respect to the $H^1$ norm with a dimension-explicit convergence rate.

A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schrödinger Eigenvalue Problems

This paper analyzes the generalization error of two-layer neural networks for computing the ground state of the Schr\"odinger operator on a $d$-dimensional hypercube. We prove that the convergence rate of the generalization error is independent of the dimension $d$, under the a priori assumption that the ground state lies in a spectral Barron space. We verify such assumption by proving a new regularity estimate for the ground state in the spectral Barron space. The later is achieved by a fixed point argument based on the Krein-Rutman theorem.

Incorporating Orientations into End-to-end Driving Model for Steering Control

In this paper, we present a novel end-to-end deep neural network model for autonomous driving that takes monocular image sequence as input, and directly generates the steering control angle. Firstly, we model the end-to-end driving problem as a local path planning process. Inspired by the environmental representation in the classical planning algorithms(i.e. the beam curvature method), pixel-wise orientations are fed into the network to learn direction-aware features. Next, to handle the imbalanced distribution of steering values in training datasets, we propose an improvement on a cost-sensitive loss function named SteeringLoss2. Besides, we also present a new end-to-end driving dataset, which provides corresponding LiDAR and image sequences, as well as standard driving behaviors. Our dataset includes multiple driving scenarios, such as urban, country, and off-road. Numerous experiments are conducted on both public available LiVi-Set and our own dataset, and the results show that the model using our proposed methods can predict steering angle accurately.

Neural Collapse with Cross-Entropy Loss

We consider the variational problem of cross-entropy loss with $n$ feature vectors on a unit hypersphere in $\mathbb{R}^d$. We prove that when $d \geq n - 1$, the global minimum is given by the simplex equiangular tight frame, which justifies the neural collapse behavior. We also prove that as $n \rightarrow \infty$ with fixed $d$, the minimizing points will distribute uniformly on the hypersphere and show a connection with the frame potential of Benedetto & Fickus.

A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations

This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of two-layer neural networks in the framework of the DRM for solving two prototype elliptic PDEs: Poisson equation and static Schr\"odinger equation on the $d$-dimensional unit hypercube. Specifically, we prove that the convergence rates of generalization errors are independent of the dimension $d$, under the a priori assumption that the exact solutions of the PDEs lie in a suitable low-complexity space called spectral Barron space. Moreover, we give sufficient conditions on the forcing term and the potential function which guarantee that the solutions are spectral Barron functions. We achieve this by developing a new solution theory for the PDEs on the spectral Barron space, which can be viewed as an analog of the classical Sobolev regularity theory for PDEs.

Complexity of zigzag sampling algorithm for strongly log-concave distributions

We study the computational complexity of zigzag sampling algorithm for strongly log-concave distributions. The zigzag process has the advantage of not requiring time discretization for implementation, and that each proposed bouncing event requires only one evaluation of partial derivative of the potential, while its convergence rate is dimension independent. Using these properties, we prove that the zigzag sampling algorithm achieves $\varepsilon$ error in chi-square divergence with a computational cost equivalent to $O\bigl(\kappa^2 d^\frac{1}{2}(\log\frac{1}{\varepsilon})^{\frac{3}{2}}\bigr)$ gradient evaluations in the regime $\kappa \ll \frac{d}{\log d}$ under a warm start assumption, where $\kappa$ is the condition number and $d$ is the dimension.

SMDS-Net: Model Guided Spectral-Spatial Network for Hyperspectral Image Denoising

Deep learning (DL) based hyperspectral images (HSIs) denoising approaches directly learn the nonlinear mapping between observed noisy images and underlying clean images. They normally do not consider the physical characteristics of HSIs, therefore making them lack of interpretability that is key to understand their denoising mechanism.. In order to tackle this problem, we introduce a novel model guided interpretable network for HSI denoising. Specifically, fully considering the spatial redundancy, spectral low-rankness and spectral-spatial properties of HSIs, we first establish a subspace based multi-dimensional sparse model. This model first projects the observed HSIs into a low-dimensional orthogonal subspace, and then represents the projected image with a multidimensional dictionary. After that, the model is unfolded into an end-to-end network named SMDS-Net whose fundamental modules are seamlessly connected with the denoising procedure and optimization of the model. This makes SMDS-Net convey clear physical meanings, i.e., learning the low-rankness and sparsity of HSIs. Finally, all key variables including dictionaries and thresholding parameters are obtained by the end-to-end training. Extensive experiments and comprehensive analysis confirm the denoising ability and interpretability of our method against the state-of-the-art HSI denoising methods.

Global optimality of softmax policy gradient with single hidden layer neural networks in the mean-field regime

We study the problem of policy optimization for infinite-horizon discounted Markov Decision Processes with softmax policy and nonlinear function approximation trained with policy gradient algorithms. We concentrate on the training dynamics in the mean-field regime, modeling e.g., the behavior of wide single hidden layer neural networks, when exploration is encouraged through entropy regularization. The dynamics of these models is established as a Wasserstein gradient flow of distributions in parameter space. We further prove global optimality of the fixed points of this dynamics under mild conditions on their initialization.

Random Coordinate Underdamped Langevin Monte Carlo

The Underdamped Langevin Monte Carlo (ULMC) is a popular Markov chain Monte Carlo sampling method. It requires the computation of the full gradient of the log-density at each iteration, an expensive operation if the dimension of the problem is high. We propose a sampling method called Random Coordinate ULMC (RC-ULMC), which selects a single coordinate at each iteration to be updated and leaves the other coordinates untouched. We investigate the computational complexity of RC-ULMC and compare it with the classical ULMC for strongly log-concave probability distributions. We show that RC-ULMC is always cheaper than the classical ULMC, with a significant cost reduction when the problem is highly skewed and high dimensional. Our complexity bound for RC-ULMC is also tight in terms of dimension dependence.