相关论文: Generating Rectifiable Measures through Neural Net…

We provide non asymptotic rates of convergence of the Wasserstein Generative Adversarial networks (WGAN) estimator. We build neural networks classes representing the generators and discriminators which yield a GAN that achieves the minimax…

We derive an approximation error bound that holds simultaneously for a function and all its derivatives up to any prescribed order. The bounds apply to elementary functions, including multivariate polynomials, the exponential function, and…

We study the necessary and sufficient complexity of ReLU neural networks---in terms of depth and number of weights---which is required for approximating classifier functions in $L^2$. As a model class, we consider the set $\mathcal{E}^\beta…

Comparing images to recommend items from an image-inventory is a subject of continued interest. Added with the scalability of deep-learning architectures the once `manual' job of hand-crafting features have been largely alleviated, and…

In this paper, we study approximation properties of single hidden layer neural networks with weights varying on finitely many directions and thresholds from an open interval. We obtain a necessary and at the same time sufficient measure…

We establish the fundamental limits in the approximation of Lipschitz functions by deep ReLU neural networks with finite-precision weights. Specifically, three regimes, namely under-, over-, and proper quantization, in terms of minimax…

Parametrized measures (or Young measures) enable to reformulate non-convex variational problems as convex problems at the cost of enlarging the search space from space of functions to space of measures. To benefit from such machinery, we…

We study the computation complexity of deep ReLU (Rectified Linear Unit) neural networks for the approximation of functions from the H\"older-Zygmund space of mixed smoothness defined on the $d$-dimensional unit cube when the dimension $d$…

We show that every $d$-dimensional probability distribution of bounded support can be generated through deep ReLU networks out of a $1$-dimensional uniform input distribution. What is more, this is possible without incurring a cost - in…

We derive rigorous bounds on the error resulting from the approximation of the solution of parametric hyperbolic scalar conservation laws with ReLU neural networks. We show that the approximation error can be made as small as desired with…

In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there…

Among the several paradigms of artificial intelligence (AI) or machine learning (ML), a remarkably successful paradigm is deep learning. Deep learning's phenomenal success has been hoped to be interpreted via fundamental research on the…

We study the robust one-bit compressed sensing problem whose goal is to design an algorithm that faithfully recovers any sparse target vector $\theta_0\in\mathbb{R}^d$ \textit{uniformly} via $m$ quantized noisy measurements. Specifically,…

Deep neural networks with rectified linear units (ReLU) are getting more and more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLU…

This paper investigates the approximation power of three types of random neural networks: (a) infinite width networks, with weights following an arbitrary distribution; (b) finite width networks obtained by subsampling the preceding…

We study the fundamental problem of ReLU regression, where the goal is to fit Rectified Linear Units (ReLUs) to data. This supervised learning task is efficiently solvable in the realizable setting, but is known to be computationally hard…

We consider the computational complexity of training depth-2 neural networks composed of rectified linear units (ReLUs). We show that, even for the case of a single ReLU, finding a set of weights that minimizes the squared error (even…

While metric and similarity learning has been extensively studied from several theoretical perspectives, a rigorous understanding of its generalization performance is still lacking. In this paper, we investigate the generalization behavior…

These series of notes serve as an introduction to some of both the classical and modern techniques in Reifenberg theory. At its heart, Reifenberg theory is about studying general sets or measures which can be, in one sense or another,…

We present a method for computing exact reachable sets for deep neural networks with rectified linear unit (ReLU) activation. Our method is well-suited for use in rigorous safety analysis of robotic perception and control systems with deep…