Federico Bassetti

Federico Bassetti, Vassili De Palma, Lucia Ladelli

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.

Federico Bassetti, Lucia Ladelli, Pietro Rotondo

We study the distributional properties of linear neural networks with random parameters in the context of large networks, where the number of layers diverges in proportion to the number of neurons per layer. Prior works have shown that in the infinite-width regime, where the number of neurons per layer grows to infinity while the depth remains fixed, neural networks converge to a Gaussian process, known as the Neural Network Gaussian Process. However, this Gaussian limit sacrifices descriptive power, as it lacks the ability to learn dependent features and produce output correlations that reflect observed labels. Motivated by these limitations, we explore the joint proportional limit in which both depth and width diverge but maintain a constant ratio, yielding a non-Gaussian distribution that retains correlations between outputs. Our contribution extends previous works by rigorously characterizing, for linear activation functions, the limiting distribution as a nontrivial mixture of Gaussians.

Federico Bassetti, Marco Gherardi, Alessandro Ingrosso et al.

Deep linear networks have been extensively studied, as they provide simplified models of deep learning. However, little is known in the case of finite-width architectures with multiple outputs and convolutional layers. In this manuscript, we provide rigorous results for the statistics of functions implemented by the aforementioned class of networks, thus moving closer to a complete characterization of feature learning in the Bayesian setting. Our results include: (i) an exact and elementary non-asymptotic integral representation for the joint prior distribution over the outputs, given in terms of a mixture of Gaussians; (ii) an analytical formula for the posterior distribution in the case of squared error loss function (Gaussian likelihood); (iii) a quantitative description of the feature learning infinite-width regime, using large deviation theory. From a physical perspective, deep architectures with multiple outputs or convolutional layers represent different manifestations of kernel shape renormalization, and our work provides a dictionary that translates this physics intuition and terminology into rigorous Bayesian statistics.

Gennaro Auricchio, Federico Bassetti, Stefano Gualandi et al.

This paper presents a novel method to compute the exact Kantorovich-Wasserstein distance between a pair of $d$-dimensional histograms having $n$ bins each. We prove that this problem is equivalent to an uncapacitated minimum cost flow problem on a $(d+1)$-partite graph with $(d+1)n$ nodes and $dn^{\frac{d+1}{d}}$ arcs, whenever the cost is separable along the principal $d$-dimensional directions. We show numerically the benefits of our approach by computing the Kantorovich-Wasserstein distance of order 2 among two sets of instances: gray scale images and $d$-dimensional biomedical histograms. On these types of instances, our approach is competitive with state-of-the-art optimal transport algorithms.

Federico Bassetti, Stefano Gualandi, Marco Veneroni

In this work, we present a method to compute the Kantorovich-Wasserstein distance of order one between a pair of two-dimensional histograms. Recent works in Computer Vision and Machine Learning have shown the benefits of measuring Wasserstein distances of order one between histograms with $n$ bins, by solving a classical transportation problem on very large complete bipartite graphs with $n$ nodes and $n^2$ edges. The main contribution of our work is to approximate the original transportation problem by an uncapacitated min cost flow problem on a reduced flow network of size $O(n)$ that exploits the geometric structure of the cost function. More precisely, when the distance among the bin centers is measured with the 1-norm or the $\infty$-norm, our approach provides an optimal solution. When the distance among bins is measured with the 2-norm: (i) we derive a quantitative estimate on the error between optimal and approximate solution; (ii) given the error, we construct a reduced flow network of size $O(n)$. We numerically show the benefits of our approach by computing Wasserstein distances of order one on a set of grey scale images used as benchmark in the literature. We show how our approach scales with the size of the images with 1-norm, 2-norm and $\infty$-norm ground distances, and we compare it with other two methods which are largely used in the literature.