arXiv subjects

Pietro Rotondo

Publications and source records attributed to Pietro Rotondo.

At least 19 recordsLinked to original sources

arXiv2026

Climate change and human mobility will shape dengue emergence risk in Europe

The risk of local arbovirus outbreaks in Europe is expected to increase due to climate change, as suggested by the multiplication of arbovirus outbreaks in the last decades. Europe has historically been a non-endemic region, making it vital to pinpoint which populations are potentially exposed -and under which conditions- so we can build truly robust epidemic preparedness capabilities. We introduce an integrated, multi-scale model that fuses a mechanistic transmission engine with a vector abundance framework, all embedded in a mobility-driven metapopulation system capturing human, vector, and air-traffic movement. To this end, we combine climate and population projections with mobility data to estimate and map dengue emergence risk in Europe throughout the 21st century. Additionally, we introduce a dedicated migration model that explores how climate-driven population redistribution could alter these risk estimates.Assuming the climate avoids major tipping points, model-derived risk indicators increase substantially under most emissions scenarios. While the spatio-temporal risk will remain largely driven by importation, our results indicate a gradual transition toward an environment-driven regime, particularly under the worst-case emissions scenario. To better anticipate and manage recurrent arbovirus outbreaks, our findings highlight the need to integrate mobility pathways and climate-driven population redistribution into predictive models of vector-borne disease emergence in temperate regions.

physics.soc-ph↗

arXiv2026

Kernel Renormalization in Bayesian Deep Neural Networks: the Equivalent Wishart Ansatz in the Proportional Regime

The scaling limit where both the size of the training set $P$ and the width $N$ of a deep neural network grow at the same rate, the so-called proportional-width regime, has been intensely studied for shallow, single-hidden-layer networks. However, extending these non-perturbative results from shallow architectures to deep non-linear networks has proven very challenging. Here we present an effective approximate approach to predict the generalization performance of Bayesian multi-layer perceptrons (MLPs) of fixed depth $L$ on arbitrary high-dimensional data. We propose an equivalent Wishart Ansatz to capture the dominant stochastic fluctuations of the hierarchical empirical kernels of MLPs. This allows us to perform a large deviation analysis for the partition function of MLPs in the proportional limit, expressed in terms of a renormalized NNGP kernel. In this description, even strong representation learning in the proportional limit is encoded in at most $L$ scalar order parameters, determined self-consistently. Extending the approach to convolutional architectures (CNNs), we identify a hierarchical local kernel renormalization mechanism, which allows to quantify more complex data-dependent transformations of the large-width kernel in CNNs due to finite-width effects. We test our effective theory against sampling experiments from the Bayesian posterior of finite deep neural networks with depths $L \sim O(10)$ and $P\sim O(10^3)$ on classic benchmark datasets, finding overall very good agreement together with two distinct types of systematic deviations.

cs.LG↗

arXiv2025

Microscopic and collective signatures of feature learning in neural networks

Feature extraction - the ability to identify relevant properties of data - is a key factor underlying the success of deep learning. Yet, it has proved difficult to elucidate its nature within existing predictive theories, to the extent that there is no consensus on the very definition of feature learning. A promising hint in this direction comes from previous phenomenological observations of quasi-universal aspects in the training dynamics of neural networks, displayed by simple properties of feature geometry. We address this problem within a statistical-mechanics framework for Bayesian learning in one hidden layer neural networks with standard parameterization. Analytical computations in the proportional limit (when both the network width and the size of the training set are large) can quantify fingerprints of feature learning, both collective ones (related to manifold geometry) and microscopic ones (related to the weights). In particular, (i) the distance between different class manifolds in feature space is a nonmonotonic function of the temperature, which we interpret as the equilibrium counterpart of a phenomenon observed under gradient descent (GD) dynamics, and (ii) the microscopic learnable parameters in the network undergo a finite data-dependent displacement with respect to the infinite-width limit, and develop correlations. These results indicate that nontrivial feature learning is at play in a regime where the posterior predictive distribution is that of Gaussian process regression with a trivially rescaled prior.

cond-mat.dis-nn↗

arXiv2024

Proportional infinite-width infinite-depth limit for deep linear neural networks

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.

stat.ML↗

arXiv2024

Statistical mechanics of transfer learning in fully-connected networks in the proportional limit

Transfer learning (TL) is a well-established machine learning technique to boost the generalization performance on a specific (target) task using information gained from a related (source) task, and it crucially depends on the ability of a network to learn useful features. Leveraging recent analytical progress in the proportional regime of deep learning theory (i.e. the limit where the size of the training set $P$ and the size of the hidden layers $N$ are taken to infinity keeping their ratio $\alpha = P/N$ finite), in this work we develop a novel single-instance Franz-Parisi formalism that yields an effective theory for TL in fully-connected neural networks. Unlike the (lazy-training) infinite-width limit, where TL is ineffective, we demonstrate that in the proportional limit TL occurs due to a renormalized source-target kernel that quantifies their relatedness and determines whether TL is beneficial for generalization.

cond-mat.dis-nn↗

arXiv2024

Feature learning in finite-width Bayesian deep linear networks with multiple outputs and convolutional layers

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.

stat.ML↗

arXiv2023

Resolution of similar patterns in a solvable model of unsupervised deep learning with structured data

Empirical data, on which deep learning relies, has substantial internal structure, yet prevailing theories often disregard this aspect. Recent research has led to the definition of structured data ensembles, aimed at equipping established theoretical frameworks with interpretable structural elements, a pursuit that aligns with the broader objectives of spin glass theory. We consider a one-parameter structured ensemble where data consists of correlated pairs of patterns, and a simplified model of unsupervised learning, whereby the internal representation of the training set is fixed at each layer. A mean field solution of the model identifies a set of layer-wise recurrence equations for the overlaps between the internal representations of an unseen input and of the training set. The bifurcation diagram of this discrete-time dynamics is topologically inequivalent to the unstructured one, and displays transitions between different phases, selected by varying the load (the number of training pairs divided by the width of the network). The network's ability to resolve different patterns undergoes a discontinuous transition to a phase where signal processing along the layers dissipates differential information about an input's proximity to the different patterns in a pair. A critical value of the parameter tuning the correlations separates regimes where data structure improves or hampers the identification of a given pair of patterns.

cond-mat.dis-nn↗

arXiv2023

Inversion dynamics of class manifolds in deep learning reveals tradeoffs underlying generalisation

To achieve near-zero training error in a classification problem, the layers of a feed-forward network have to disentangle the manifolds of data points with different labels, to facilitate the discrimination. However, excessive class separation can bring to overfitting since good generalisation requires learning invariant features, which involve some level of entanglement. We report on numerical experiments showing how the optimisation dynamics finds representations that balance these opposing tendencies with a non-monotonic trend. After a fast segregation phase, a slower rearrangement (conserved across data sets and architectures) increases the class entanglement.The training error at the inversion is stable under subsampling, and across network initialisations and optimisers, which characterises it as a property solely of the data structure and (very weakly) of the architecture. The inversion is the manifestation of tradeoffs elicited by well-defined and maximally stable elements of the training set, coined ``stragglers'', particularly influential for generalisation.

cs.LG↗

arXiv2021

Self-induced glassy phase in multimodal cavity quantum electrodynamics

We provide strong evidence that the effective spin-spin interaction in a multimodal confocal optical cavity gives rise to a self-induced glassy phase, which emerges exclusively from the peculiar euclidean correlations and is not related to the presence of disorder as in standard spin glasses. As recently shown, this spin-spin effective interaction is both non-local and non-translational invariant, and randomness in the atoms positions produces a spin glass phase. Here we consider the simplest feasible disorder-free setting where atoms form a one-dimensional regular chain and we study the thermodynamics of the resulting effective Ising model. We present extensive results showing that the system has a low-temperature glassy phase. Notably, for rational values of the only free adimensional parameter $\alpha=p/q$ of the interaction, the number of metastable states at low temperature grows exponentially with $q$ and the problem of finding the ground state rapidly becomes computationally intractable, suggesting that the system develops high energy barriers and ergodicity breaking occurs.

cond-mat.stat-mech↗

arXiv2020

Random geometric graphs in high dimension

Many machine learning algorithms used for dimensional reduction and manifold learning leverage on the computation of the nearest neighbours to each point of a dataset to perform their tasks. These proximity relations define a so-called geometric graph, where two nodes are linked if they are sufficiently close to each other. Random geometric graphs, where the positions of nodes are randomly generated in a subset of $\mathbb{R}^{d}$, offer a null model to study typical properties of datasets and of machine learning algorithms. Up to now, most of the literature focused on the characterization of low-dimensional random geometric graphs whereas typical datasets of interest in machine learning live in high-dimensional spaces ($d \gg 10^{2}$). In this work, we consider the infinite dimensions limit of hard and soft random geometric graphs and we show how to compute the average number of subgraphs of given finite size $k$, e.g. the average number of $k$-cliques. This analysis highlights that local observables display different behaviors depending on the chosen ensemble: soft random geometric graphs with continuous activation functions converge to the naive infinite dimensional limit provided by Erdös-Rényi graphs, whereas hard random geometric graphs can show systematic deviations from it. We present numerical evidence that our analytical insights, exact in infinite dimensions, provide a good approximation also for dimension $d\gtrsim10$.

cond-mat.stat-mech↗

arXiv2020

Statistical learning theory of structured data

The traditional approach of statistical physics to supervised learning routinely assumes unrealistic generative models for the data: usually inputs are independent random variables, uncorrelated with their labels. Only recently, statistical physicists started to explore more complex forms of data, such as equally-labelled points lying on (possibly low dimensional) object manifolds. Here we provide a bridge between this recently-established research area and the framework of statistical learning theory, a branch of mathematics devoted to inference in machine learning. The overarching motivation is the inadequacy of the classic rigorous results in explaining the remarkable generalization properties of deep learning. We propose a way to integrate physical models of data into statistical learning theory, and address, with both combinatorial and statistical mechanics methods, the computation of the Vapnik-Chervonenkis entropy, which counts the number of different binary classifications compatible with the loss class. As a proof of concept, we focus on kernel machines and on two simple realizations of data structure introduced in recent physics literature: $k$-dimensional simplexes with prescribed geometric relations and spherical manifolds (equivalent to margin classification). Entropy, contrary to what happens for unstructured data, is nonmonotonic in the sample size, in contrast with the rigorous bounds. Moreover, data structure induces a novel transition beyond the storage capacity, which we advocate as a proxy of the nonmonotonicity, and ultimately a cue of low generalization error. The identification of a synaptic volume vanishing at the transition allows a quantification of the impact of data structure within replica theory, applicable in cases where combinatorial methods are not available, as we demonstrate for margin learning.

cond-mat.stat-mech↗

arXiv2020

Beyond the storage capacity: data driven satisfiability transition

Data structure has a dramatic impact on the properties of neural networks, yet its significance in the established theoretical frameworks is poorly understood. Here we compute the Vapnik-Chervonenkis entropy of a kernel machine operating on data grouped into equally labelled subsets. At variance with the unstructured scenario, entropy is non-monotonic in the size of the training set, and displays an additional critical point besides the storage capacity. Remarkably, the same behavior occurs in margin classifiers even with randomly labelled data, as is elucidated by identifying the synaptic volume encoding the transition. These findings reveal aspects of expressivity lying beyond the condensed description provided by the storage capacity, and they indicate the path towards more realistic bounds for the generalization error of neural networks.

cs.LG↗

arXiv2020

Intrinsic dimension estimation for locally undersampled data

High-dimensional data are ubiquitous in contemporary science and finding methods to compress them is one of the primary goals of machine learning. Given a dataset lying in a high-dimensional space (in principle hundreds to several thousands of dimensions), it is often useful to project it onto a lower-dimensional manifold, without loss of information. Identifying the minimal dimension of such manifold is a challenging problem known in the literature as intrinsic dimension estimation (IDE). Traditionally, most IDE algorithms are either based on multiscale principal component analysis (PCA) or on the notion of correlation dimension (and more in general on k-nearest-neighbors distances). These methods are affected, in different ways, by a severe curse of dimensionality. In particular, none of the existing algorithms can provide accurate ID estimates in the extreme locally undersampled regime, i.e. in the limit where the number of samples in any local patch of the manifold is less than (or of the same order of) the ID of the dataset. Here we introduce a new ID estimator that leverages on simple properties of the tangent space of a manifold to overcome these shortcomings. The method is based on the full correlation integral, going beyond the limit of small radius used for the estimation of the correlation dimension. Our estimator alleviates the extreme undersampling problem, intractable with other methods. Based on this insight, we explore a multiscale generalization of the algorithm. We show that it is capable of (i) identifying multiple dimensionalities in a dataset, and (ii) providing accurate estimates of the ID of extremely curved manifolds. In particular, we test the method on manifolds generated from global transformations of high-contrast images, relevant for invariant object recognition and considered a challenge for state-of-the-art ID estimators.

cs.LG↗

arXiv2020

Signatures of associative memory behavior in a multi-mode spin-boson model

Spin-boson models can describe a variety of physical systems, such as atoms in a cavity or vibrating ion chains. In equilibrium these systems often feature a radical change in their behavior when switching from weak to strong spin-boson interaction. This usually manifests in a transition from a "dark" to a "superradiant" phase. However, understanding the out-of-equilibrium physics of these models is extremely challenging, and even more so for strong spin-boson coupling. Here we show that non-equilibrium strongly interacting spin-boson systems can mimic some fundamental properties of an associative memory - a system which permits the recognition of patterns, such as letters of an alphabet. Patterns are encoded in the couplings between spins and bosons, and we discuss the dynamics of the spins from the perspective of pattern retrieval in associative memory models. We identify two phases, a "paramagnetic" and a "ferromagnetic" one, and a crossover behavior between these regimes. The "ferromagnetic" phase is reminiscent of pattern retrieval. We highlight similarities and differences with the thermal dynamics of a Hopfield associative memory and show that indeed elements of "machine learning behavior" emerge in strongly coupled spin-boson systems.

quant-ph↗

arXiv2019

Large deviations of the free energy in the p-spin glass spherical model

We investigate the behavior of the rare fluctuations of the free energy in the p-spin spherical model, evaluating the corresponding rate function via the Gärtner-Ellis theorem. This approach requires the knowledge of the analytic continuation of the disorder-averaged replicated partition function to arbitrary real number of replicas. In zero external magnetic field, we show via a one-step replica symmetry breaking (1RSB) calculation that the rate function is infinite for fluctuations of the free energy above its typical value, corresponding to an anomalous, super-extensive suppression of rare fluctuations. We extend this calculation to non-zero magnetic field, showing that in this case this very large deviation disappears and we try to motivate this finding in light of a geometrical interpretation of the scaled cumulant generating function.

cond-mat.dis-nn↗

arXiv2019

Dynamics of strongly coupled disordered dissipative spin-boson systems

Spin-boson Hamiltonians are an effective description for numerous quantum many-body systems such as atoms coupled to cavity modes, quantum electrodynamics in circuits and trapped ion systems. While reaching the limit of strong coupling is possible in current experiments, the understanding of the physics in this parameter regime remains a challenge, especially when disorder and dissipation are taken into account. Here we investigate a regime where the many-body spin dynamics can be related to a Ising energy function defined in terms of the spin-boson couplings. While in the coherent weak coupling regime it is known that an effective description in terms of spin Hamiltonian is possible, we show that a similar viewpoint can be adopted in the presence of dissipation and strong couplings. The resulting many-body dynamics features approximately thermal regimes, separated by out-of-equilibrium ones in which detailed balance is broken. Moreover, we show that under appropriately chosen conditions one can even achieve cooling of the spin degrees of freedom. This points towards the possibility of using strongly coupled dissipative spin-boson systems for engineering complex energy landscapes together with an appropriate cooling dynamics.

cond-mat.stat-mech↗

arXiv2019

Counting the learnable functions of structured data

Cover's function counting theorem is a milestone in the theory of artificial neural networks. It provides an answer to the fundamental question of determining how many binary assignments (dichotomies) of $p$ points in $n$ dimensions can be linearly realized. Regrettably, it has proved hard to extend the same approach to more advanced problems than the classification of points. In particular, an emerging necessity is to find methods to deal with structured data, and specifically with non-pointlike patterns. A prominent case is that of invariant recognition, whereby identification of a stimulus is insensitive to irrelevant transformations on the inputs (such as rotations or changes in perspective in an image). An object is therefore represented by an extended perceptual manifold, consisting of inputs that are classified similarly. Here, we develop a function counting theory for structured data of this kind, by extending Cover's combinatorial technique, and we derive analytical expressions for the average number of dichotomies of generically correlated sets of patterns. As an application, we obtain a closed formula for the capacity of a binary classifier trained to distinguish general polytopes of any dimension. These results may help extend our theoretical understanding of generalization, feature extraction, and invariant object recognition by neural networks.

cond-mat.dis-nn↗

arXiv2019

Generalization from correlated sets of patterns in the perceptron

Generalization is a central aspect of learning theory. Here, we propose a framework that explores an auxiliary task-dependent notion of generalization, and attempts to quantitatively answer the following question: given two sets of patterns with a given degree of dissimilarity, how easily will a network be able to "unify" their interpretation? This is quantified by the volume of the configurations of synaptic weights that classify the two sets in a similar manner. To show the applicability of our idea in a concrete setting, we compute this quantity for the perceptron, a simple binary classifier, using the classical statistical physics approach in the replica-symmetric ansatz. In this case, we show how an analytical expression measures the "distance-based capacity", the maximum load of patterns sustainable by the network, at fixed dissimilarity between patterns and fixed allowed number of errors. This curve indicates that generalization is possible at any distance, but with decreasing capacity. We propose that a distance-based definition of generalization may be useful in numerical experiments with real-world neural networks, and to explore computationally sub-dominant sets of synaptic solutions.

cond-mat.dis-nn↗