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Let $W_{\infty}(\beta)$ be the limit of the Biggins martingale $W_n(\beta)$ associated to a supercritical branching random walk with mean number of offspring $m$. We prove a functional central limit theorem stating that as $n\to\infty$ the…

Leveraging the infinite dimensional neural network architecture we proposed in arXiv:2109.13512v4 and which can process inputs from Fr\'echet spaces, and using the universal approximation property shown therein, we now largely extend the…

The dynamics of neural networks is often characterized by collective behavior and quasi-synchronous events, where a large fraction of neurons fire in short time intervals, separated by uncorrelated firing activity. These global temporal…

We consider Berry's random planar wave model (1977), and prove spatial functional limit theorems - in the high-energy limit - for discretized and truncated versions of the random field obtained by restricting its nodal length to rectangular…

Many practical problems need the output of a machine learning model to satisfy a set of constraints, $K$. Nevertheless, there is no known guarantee that classical neural network architectures can exactly encode constraints while…

We construct a family of non-parametric (infinite-dimensional) manifolds of finite measures on $R^d$. The manifolds are modelled on a variety of weighted Sobolev spaces, including Hilbert-Sobolev spaces and mixed-norm spaces. Each supports…

We investigate the realizations of a random Gaussian field on a finite domain of ${\mathbb R}^d$ in the limit where a given linear functional of the field is large. We prove that if its variance is bounded, the field converges uniformly and…

By using the $\tau$-topology of Kryszewski and Szulkin, we establish a natural new version of the Saddle Theorem for strongly indefinite functionals. The abstract result will be applied for studying the existence of a nontrivial solution of…

We consider a new approach in the definition of two-dimensional heavy-tailed distributions. Namely, we introduce the classes of two-dimensional long-tailed, of twodimensional dominatedly varying and of two-dimensional consistently varying…

We consider an inhomogeneous Erd\H{o}s-R\'enyi random graph ensemble with exponentially decaying random disconnection probabilities determined by an i.i.d. field of variables with heavy tails and infinite mean associated to the vertices of…

We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the…

This paper studies the numerical approximation of divergence-free vector fields by linearized shallow neural networks, also referred to as random feature models or finite neuron spaces. Combining the stable potential lifting for…

Despite the practical success of deep neural networks, a comprehensive theoretical framework that can predict practically relevant scores, such as the test accuracy, from knowledge of the training data is currently lacking. Huge…

In variational inference, the benefits of Bayesian models rely on accurately capturing the true posterior distribution. We propose using neural samplers that specify implicit distributions, which are well-suited for approximating complex…

A simple way of obtaining robust estimates of the "center" (or the "location") and of the "scatter" of a dataset is to use the maximum likelihood estimate with a class of heavy-tailed distributions, regardless of the "true" distribution…

From the classical and influential works of Neal (1996), it is known that the infinite width scaling limit of a Bayesian neural network with one hidden layer is a Gaussian process, when the network weights have bounded prior variance.…

The Bayesian brain hypothesis postulates that the brain accurately operates on statistical distributions according to Bayes' theorem. The random failure of presynaptic vesicles to release neurotransmitters may allow the brain to sample from…

We consider the problem of random sampling for band-limited functions. When can a band-limited function $f$ be recovered from randomly chosen samples $f(x_j), j\in \mathbb{N}$? We estimate the probability that a sampling inequality of the…

We show that deep neural networks trained across diverse tasks exhibit remarkably similar low-dimensional parametric subspaces. We provide the first large-scale empirical evidence that demonstrates that neural networks systematically…

We study the profile $X_{n,k}$ of random search trees including binary search trees and $m$-ary search trees. Our main result is a functional limit theorem of the normalized profile $X_{n,k}/\mathbb{E}X_{n,k}$ for $k=\lfloor\alpha\log…