Vassili De Palma | Search arXiv

arXiv2026

Gaussian equivalence for CNN with non-Gaussian weights

We study the infinite-width limit of convolutional neural networks with random weights and show that the Gaussian process limit does not require Gaussian priors. Assuming that the weights are i.i.d., centered, and have sufficiently many finite moments, converges weakly in the space of continuous functions on the compact input set to independent copies of a centered Gaussian process. Our framework accommodates activation functions of polynomial growth together with a broad class of patch-extraction operators, covering standard convolutions, padding, stride, and common pooling mechanisms.

math.PR↗

arXiv2026

Large deviation principles for convolutional Bayesian neural networks

While suitably scaled CNNs with Gaussian initialization are known to converge to Gaussian processes as the number of channels diverges, little is known beyond this Gaussian limit. We establish a large deviation principle (LDP) for convolutional neural networks in the infinite-channel regime. We consider a broad class of multidimensional CNN architectures characterized by general receptive fields encoded through a patch-extractor function satisfying mild structural assumptions. Our main result establishes a large deviation principle (LDP) for the sequence of conditional covariance matrices under Gaussian prior distribution on the weights. We further derive an LDP for the posterior distribution obtained by conditioning on a finite number of observations. In addition, we provide a streamlined proof of the concentration of the conditional covariances and of the Gaussian equivalence of the network. To the best of our knowledge, this is the first large deviation principle established for convolutional neural networks.

math.PR↗