Adriano Barra

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• Professor
• Professor (Associate) at Sapienza University of Rome

By leveraging tools from the statistical mechanics of complex systems, in these short notes we extend the architecture of a neural network for hetero-associative memory (called three-directional associative memories, TAM) to explore supervised and unsupervised learning protocols. In particular, by providing entropic-heterogeneous datasets to its va...

Auto-associative neural networks (e.g. the Hopfield model implementing the standard Hebbian prescription) serve as a foundational framework for pattern recognition and associative memory in statistical mechanics. However, their hetero-associative counterparts, though less explored, exhibit even richer computational capabilities. In this work, we ex...

The common thread behind the recent Nobel Prize in Physics to John Hopfield and those conferred to Giorgio Parisi in 2021 and Philip Anderson in 1977 is disorder. Quoting Philip Anderson: "more is different". This principle has been extensively demonstrated in magnetic systems and spin glasses, and, in this work, we test its validity on Hopfield ne...

Recently, the original storage prescription for the Hopfield model of neural networks – as well as for its dense generalizations – has been turned into a genuine Hebbian learning rule by postulating the expression of its Hamiltonian for both the supervised and unsupervised protocols. In these notes, first, we obtain these explicit expressions by re...

In the last five decades, mean-field neural-networks have played a crucial role in modelling associative memories and, in particular, the Hopfield model has been extensively studied using tools borrowed from the statistical mechanics of spin glasses. However, achieving mathematical control of the infinite-volume limit of the model's free-energy has...

While auto-associative neural networks (e.g., the Hopfield model implementing the standard Hebbian prescription for learning) play as the reference setting for pattern recognition and associative memory in statistical mechanics, hetero-associative extensions (despite much less investigated) display richer emergent computational skills. Here we stud...

Pyramidal cells that emit spikes when the animal is at specific locations of the environment are known as "place cells": these neurons are thought to provide an internal representation of space via "cognitive maps". Here, we consider the Battaglia-Treves neural network model for cognitive map storage and reconstruction, instantiated with McCulloch...

Although instantaneous interactions are unphysical, a large variety of maximum entropy statistical inference methods match the model-inferred and the empirically measured equal-time correlation functions. Focusing on collective motion of active units, this constraint is reasonable when the interaction timescale is much faster than that of the inter...

Hebbian neural networks with multi-node interactions, often called Dense Associative Memories, have recently attracted considerable interest in the statistical mechanics community, as they have been shown to outperform their pairwise counterparts in a number of features, including resilience against adversarial attacks, pattern retrieval with extre...

Parallel learning , namely the simultaneous learning of multiple patterns, constitutes a modern challenge for neural networks. While this cannot be accomplished by standard Hebbian associative neural networks, in this paper we show how the multitasking Hebbian network (a variation on the theme of the Hopfield model, working on sparse datasets) is n...

Even though instantaneous interactions are unphysical, a large variety of maximum-entropy statistical-inference methods match the model inferred and the empirically measured equal-time correlation functions. While this constraint holds when the interaction timescale is much faster than that of the interacting units, as, e.g., in starling flocks (wh...

Spin-glasses constitute a well-grounded framework for evolutionary models. Of particular interest for (some of) these models is the lack of self-averaging of their order parameters (e.g., the Hamming distance between the genomes of two individuals), even in asymptotic limits, much as like what happens to the overlap between the configurations of tw...

A modern challenge of Artificial Intelligence is learning multiple patterns at once (i.e.parallel learning). While this can not be accomplished by standard Hebbian associative neural networks, in this paper we show how the Multitasking Hebbian Network (a variation on theme of the Hopfield model working on sparse data-sets) is naturally able to perf...

We study bi-directional associative neural networks that, exposed to noisy examples of an extensive number of random archetypes, learn the latter (with or without the presence of a teacher) when the supplied information is enough: in this setting, learning is heteroassociative -- involving couples of patterns -- and it is achieved by reverberating...

Spin-glasses constitute a well-grounded framework for evolutionary models. Of particular interest for (some of) these models is the lack of self-averaging of their order parameters (e.g. the Hamming distance between the genomes of two individuals), even in asymptotic limits, much as like the behavior of the overlap between the configurations of two...

In this paper we investigate the equilibrium properties of bidirectional associative memories (BAMs). Introduced by Kosko in 1988 as a generalization of the Hopfield model to a bipartite structure, the simplest architecture is defined by two layers of neurons, with synaptic connections only between units of different layers: even without internal c...

Hebb's learning traces its origin in Pavlov's classical conditioning; however, while the former has been extensively modeled in the past decades (e.g., by the Hopfield model and countless variations on theme), as for the latter, modeling has remained largely unaddressed so far. Furthermore, a mathematical bridge connecting these two pillars is tota...

In this work we present a rigorous and straightforward method to detect the onset of the instability of replica-symmetric theories in information processing systems, which does not require a full replica analysis as in the method originally proposed by Almeida-Thouless for spin glasses. The method is based on an expansion of the free-energy obtaine...

A crucial challenge in medicine is choosing which drug (or combination) will be the most advantageous for a particular patient. Usually, drug response rates differ substantially, and the reasons for this response unpredictability remain ambiguous. Consequently, it is central to classify features that contribute to the observed drug response variabi...

Fluorescent pH-sensor nanofiber scaffolds with constraint-based inverse modeling allow for a non-invasive spatial metabolic flux analysis able to resolve fermentation flux at single-cell resolution and the ensuing intercellular interactions in complex cellular systems. View the article: https://doi.org/10.1021/acsnano.2c06114

In neural network's Literature, "Hebbian learning" traditionally refers to the procedure by which the Hopfield model and its generalizations "store" archetypes (i.e., definite patterns that are experienced just once to form the synaptic matrix). However, the term "learning" in Machine Learning refers to the ability of the machine to extract feature...

The homeostatic control of their environment is an essential task of living cells. It has been hypothesized that, when microenvironmental pH inhomogeneities are induced by high cellular metabolic activity, diffusing protons act as signaling molecules, driving the establishment of exchange networks sustained by the cell-to-cell shuttling of overflow...

We consider dense, associative neural-networks trained with no supervision and we investigate their computational capabilities analytically, via a statistical-mechanics approach, and numerically, via Monte Carlo simulations. In particular, we obtain a phase diagram summarizing their performance as a function of the control parameters such as the qu...

We consider dense, associative neural-networks trained by a teacher (i.e., with supervision) and we investigate their computational capabilities analytically, via statistical-mechanics of spin glasses, and numerically, via Monte Carlo simulations. In particular, we obtain a phase diagram summarizing their performance as a function of the control pa...

In this paper we investigate the equilibrium properties of bidirectional associative memories (BAMs). Introduced by Kosko in 1988 as a generalization of the Hopfield model to a bipartite structure, the simplest architecture is defined by two layers of neurons, with synaptic connections only between units of different layers: even without internal c...

Understanding the glassy nature of neural networks is pivotal both for theoretical and computational advances in Machine Learning and Theoretical Artificial Intelligence. Keeping the focus on dense associative Hebbian neural networks (i.e. Hopfield networks with polynomial interactions of even degree \(P >2\)), the purpose of this paper is twofold:...

As well known, Hebb's learning traces its origin in Pavlov's Classical Conditioning, however, while the former has been extensively modelled in the past decades (e.g., by Hopfield model and countless variations on theme), as for the latter modelling has remained largely unaddressed so far; further, a bridge between these two pillars is totally lack...

Inspired by a formal equivalence between the Hopfield model and restricted Boltzmann machines (RBMs), we design a Boltzmann machine, referred to as the dreaming Boltzmann machine (DBM), which achieves better performances than the standard one. The novelty in our model lies in a precise prescription for intralayer connections among hidden neurons wh...

The homeostatic control of their environment is an essential task of living cells. It has been hypothesized that when microenvironmental pH inhomo-geneities are induced by high cellular metabolic activity, diffusing protons act as signaling molecules, driving the establishment of cross-feeding networks sustained by the cell-to-cell shuttling of ove...

The work focuses on high-performing optical ratiometric pH sensors based on fluorescent spherical microparticles. Using a fully and rapid automated computational approach, the organelle acidification dynamics of living tumor cells are precisely and noninvasively measured by tracking the position of the sensors and quantifying their response to the...

The gap between the huge volumes of data needed to train artificial neural networks and the relatively small amount of data needed by their biological counterparts is a central puzzle in machine learning. Here, inspired by biological information-processing, we introduce a generalized Hopfield network where pairwise couplings between neurons are bui...

pH balance and regulation within organelles are fundamental to cell homeostasis and proliferation. The ability to track pH in cells becomes significantly important to understand these processes in detail. Fluorescent sensors based on micro- and nanoparticles have been applied to measure intracellular pH; however, an accurate methodology to precisel...

We consider restricted Boltzmann machine (RBMs) trained over an unstructured dataset made of blurred copies of definite but unavailable “archetypes” and we show that there exists a critical sample size beyond which the RBM can learn archetypes, namely the machine can successfully play as a generative model or as a classifier, according to the opera...

In neural network's Literature, {\em Hebbian learning} traditionally refers to the procedure by which the Hopfield model and its generalizations {\em store} archetypes (i.e., definite patterns that are experienced just once to form the synaptic matrix). However, the term {\em learning} in Machine Learning refers to the ability of the machine to ext...

Understanding the glassy nature of neural networks is pivotal both for theoretical and computational advances in Machine Learning and Theoretical Artificial Intelligence. Keeping the focus on dense associative Hebbian neural networks, the purpose of this paper is two-fold: at first we develop rigorous mathematical approaches to address properly a s...

We consider restricted Boltzmann machine (RBMs) trained over an unstructured dataset made of blurred copies of definite but unavailable ``archetypes'' and we show that there exists a critical sample size beyond which the RBM can learn archetypes, namely the machine can successfully play as a generative model or as a classifier, according to the ope...

Restricted Boltzmann machines (RBMs) with a binary visible layer of size N and a Gaussian hidden layer of size P have been proved to be equivalent to a Hopfield neural network (HNN) made of N binary neurons and storing P patterns ξ, as long as the weights w in the former are identified with the patterns. Here we aim to leverage this equivalence to...

Organ-on-a-chip (OoCs) platforms could revolutionize drug discovery and might ultimately become essential tools for precision therapy. Although many single-organ and interconnected systems have been described, the immune system has been comparatively neglected, despite its pervasive role in the body and the trend towards newer therapeutic products...

In this work we apply statistical mechanics tools to infer cardiac pathologies over a sample of M patients whose heart rate variability has been recorded via 24 h Holter device and that are divided in different classes according to their clinical status (providing a repository of labelled data). Considering the set of inter-beat interval sequences...

In this paper we adapt the broken replica interpolation technique (developed by Francesco Guerra to deal with the Sherrington-Kirkpatrick model, namely a pairwise mean-field spin-glass whose couplings are i.i.d. standard Gaussian variables) in order to work also with the Hopfield model (i.e. a pairwise mean-field neural-network whose couplings are...

In this paper we study the properties of the quenched pressure of a multi-layer spin-glass model (a deep Boltzmann Machine in artificial intelligence jargon) whose pairwise interactions are allowed between spins lying in adjacent layers and not inside the same layer nor among layers at distance larger than one. We prove a theorem that bounds the qu...

While 3D cell cultures have been used as suitable representatives of in vivo conditions compared to 2D systems, scalability and flexibility in designing such platforms has been a major challenge¹. Micron-sized 3D culture platforms offer the possibility of high throughput analysis, however major challenges exist with respect to their overall design....

In this paper we develop statistical algorithms to infer possible cardiac pathologies, based on data collected from 24 h Holter recording over a sample of 2829 labelled patients; labels highlight whether a patient is suffering from cardiac pathologies. In the first part of the work we analyze statistically the heart-beat series associated to each p...

Background: In cancer research, studying cell motility is fundamental to investigate cancer invasion and drug resistance. In solid tumors constituted by a huge stroma component, such as pancreatic ductal adenocarcinoma (PDAC), the ability to quantify cell movement and interactions is mandatory to better understand the complex crosstalk between canc...

In this paper we adapt the broken replica interpolation technique (developed by Francesco Guerra to deal with the Sherrington-Kirkpatrick model, namely a pairwise mean-field spin-glass whose couplings are i.i.d. standard Gaussian variables) in order to work also with the Hopfield model (i.e., a pairwise mean-field neural-network whose couplings are...

In this work we develop analytical techniques to investigate a broad class of associative neural networks set in the high-storage regime. These techniques translate the original statistical–mechanical problem into an analytical-mechanical one which implies solving a set of partial differential equations, rather than tackling the canonical probabili...

Migration of cells can be characterized by two prototypical types of motion: individual and collective migration. We propose a statistical inference approach designed to detect the presence of cell-cell interactions that give rise to collective behaviors in cell motility experiments. This inference method has been first successfully tested on synth...

In this paper we study the properties of the quenched pressure of a multi-layer spin-glass model (a deep Boltzmann Machine in artificial intelligence jargon) whose pairwise interactions are allowed between spins lying in adjacent layers and not inside the same layer nor among layers at distance larger than one. We prove a theorem that bounds the qu...

We consider a three-layer Sejnowski machine and show that features learnt via contrastive divergence have a dual representation as patterns in a dense associative memory of order P=4. The latter is known to be able to Hebbian store an amount of patterns scaling as N^(P−1), where N denotes the number of constituting binary neurons interacting P wise...

Migration of cells can be characterized by two, prototypical types of motion: individual and collective migration. We propose a statistical-inference approach designed to detect the presence of cell-cell interactions that give rise to collective behaviors in cell-motility experiments. Such inference method has been first successfully tested on synt...

We consider a three-layer Sejnowski machine and show that features learnt via contrastive divergence have a dual representation as patterns in a dense associative memory of order P=4. The latter is known to be able to Hebbian-store an amount of patterns scaling as N^{P-1}, where N denotes the number of constituting binary neurons interacting P-wise...

In this work we develop analytical techniques to address, in the high-storage regime, phase diagrams for a broad class of neural networks (whose cost functions may include various, polynomial, contributions both in the neurons and the patterns, hence {\em dense}). The techniques translate the original statistical-mechanical problem into a pure anal...

In recent decades, statistical mechanics of disordered systems (mainly spin-glasses) has become one of the main tools to investigate complex systems, probably due to the celebrated Replica Symmetry Breaking scheme of Parisi Theory and its deep implications. In this work we consider the analog bipartite spin-glass (or real-valued restricted Boltzman...

The standard Hopfield model for associative neural networks accounts for biological Hebbian learning and acts as the harmonic oscillator for pattern recognition, however its maximal storage capacity is α∼0.14, far from the theoretical bound for symmetric networks, i.e. α=1. Inspired by sleeping and dreaming mechanisms in mammal brains, we propose a...

Restricted Boltzmann machines (RBMs) constitute one of the main models for machine statistical inference and they are widely employed in artificial intelligence as powerful tools for (deep) learning. However, in contrast with countless remarkable practical successes, their mathematical formalization has been largely elusive: from a statistical-mech...

The relativistic Hopfield model constitutes a generalization of the standard Hopfield model that is derived by the formal analogy between the statistical-mechanic framework embedding neural networks and the Lagrangian mechanics describing a fictitious single-particle motion in the space of the tuneable parameters of the network itself. In this anal...

Recently a daily routine for associative neural networks has been proposed: the network Hebbian-learns during the awake state (thus behaving as a standard Hopfield model), then, during its sleep state, optimizing information storage, it consolidates pure patterns and removes spurious ones: this forces the synaptic matrix to collapse to the projecto...

In this work we consider the {\em analog bipartite spin-glass} (or {\em real-valued restricted Boltzmann machine} in a neural network jargon), whose variables (those quenched as well as those dynamical) share standard Gaussian distributions. First, via Guerra's interpolation technique, we express its quenched free energy in terms of the natural ord...

The standard Hopfield model for associative neural networks accounts for biological Hebbian learning and acts as the harmonic oscillator for pattern recognition, however its maximal storage capacity is $\alpha \sim 0.14$, far from the theoretical bound for symmetric networks, i.e. $\alpha =1$. Inspired by sleeping and dreaming mechanisms in mammal...

The relativistic Hopfield model constitutes a generalization of the standard Hopfield model that is derived by the formal analogy between the statistical-mechanic framework embedding neural networks and the Lagrangian mechanics describing a fictitious single-particle motion in the space of the tuneable parameters of the network itself. In this anal...

Restricted Boltzmann machines (RBMs) constitute one of the main models for machine statistical inference and they are widely employed in Artificial Intelligence as powerful tools for (deep) learning. However, in contrast with countless remarkable practical successes, their mathematical formalization has been largely elusive: from a statistical-mech...

Organ-on-a-chip (OoCs) platforms could revolutionize drug discovery and might ultimately become essential tools for precision therapy. Although many single-organ and interconnected systems have been described, the immune system has been comparatively neglected, despite its pervasive role in the body and the trend towards newer therapeutic products...

We propose a modification of the cost function of the Hopfield model whose salient features shine in its Taylor expansion and result in more than pairwise interactions with alternate signs, suggesting a unified framework for handling both with deep learning and network pruning. In our analysis, we heavily rely on the Hamilton-Jacobi correspondence...

In recent years Italy has been involved in massive migration flows and, consequently, migrant integration is becoming a urgent political, economic and social issue. In this paper we apply quantitative methods, based on probability theory and statistical mechanics, to study the relative integration of migrants in Italy. In particular, we focus on th...

We study the phase diagram of a minority game where three classes of agents are present. Two types of agents play a risk-loving game that we model by the standard Snowdrift Game. The behaviour of the third type of agents is coded by {\em indifference} w.r.t. the game at all: their dynamics is designed to account for risk-aversion as an innovative b...

Complex biochemical pathways or regulatory enzyme kinetics can be reduced to chains of elementary reactions, which can be described in terms of chemical kinetics. This discipline provides a set of tools for quantifying and understanding the dialogue between reactants, whose framing into a solid and consistent mathematical description is of pivotal...

We study the phase diagram of a minority game where three classes of agents are present. Two types of agents play a risk-loving game that we model by the standard Snowdrift Game. The behaviour of the third type of agents is coded by {\em indifference} w.r.t. the game at all: their dynamics is designed to account for risk-aversion as an innovative b...

We propose a modification of the cost function of the Hopfield model whose salient features shine in its Taylor expansion and result in more than pairwise interactions with alternate signs, suggesting a unified framework for handling both with deep learning and network pruning. In our analysis, we heavily rely on the Hamilton-Jacobi correspondence...

Complex biochemical pathways or regulatory enzyme kinetics can be reduced to chains of elementary reactions, which can be described in terms of chemical kinetics. This discipline provides a set of tools for quantifying and understanding the dialogue between reactants, whose framing into a solid and consistent mathematical description is of pivotal...

In this paper we discuss the applicability of numerical descriptors and statistical physics concepts to characterize complex biological systems observed at microscopic level through organ on chip approach. To this end, we employ data collected on a microfluidic platform in which leukocytes can move through suitably built channels toward their targe...

Restricted Boltzmann Machines are key tools in Machine Learning and are described by the energy function of bipartite spin-glasses. From a statistical mechanical perspective, they share the same Gibbs measure of Hopfield networks for associative memory. In this equivalence, weights in the former play as patterns in the latter. As Boltzmann machines...

The similarity between neural and immune networks has been known for decades, but so far we did not understand the mechanism that allows the immune system, unlike associative neural networks, to recall and execute a large number of memorized defense strategies {\em in parallel}. The explanation turns out to lie in the network topology. Neurons inte...

Restricted Boltzmann Machines are key tools in Machine Learning and are described by the energy function of bipartite spin-glasses. From a statistical mechanical perspective, they share the same Gibbs measure of Hopfield networks for associative memory. In this equivalence, weights in the former play as patterns in the latter. As Boltzmann machines...