arXiv subjects

Linda Albanese

Publications and source records attributed to Linda Albanese.

17 recordsLinked to original sources

arXiv2026

Semi-supervised Hopfield model: Theoretical and Numerical results

In the daily practice of Machine Learning, fully labeled datasets are a luxury: labels demand expensive and time-consuming human annotation, whereas raw, unlabeled data can be harvested automatically and in bulk. Semi-supervised learning, where the network jointly exploits the few labeled and the many unlabeled examples at its disposal, is the standard answer to this asymmetry, yet a statistical mechanical theory of semi-supervised Hebbian learning is still lacking. In this paper we fill this gap for the Hopfield network: we prescribe a synaptic coupling given by the convex combination, weighted by a mixing parameter λin [0,1], of the supervised and unsupervised Hebbian kernels built from the same archetypes, and we solve for the emergent computational capabilities of the resulting network. A signal-to-noise analysis yields the one-step Mattis magnetization and the learning threshold, i.e. the minimum dataset size for stable retrieval. Using Guerra's interpolation, we then derive the Replica Symmetric quenched pressure in the high-storage regime, treating the correlated disorder generated by the supervised and unsupervised channels through a particular eigen-channel decomposition. The resulting phase diagram shows that a mixed strategy outperforms both pure protocols. Finally, we prove that the quenched pressure is convex in λ, so thermodynamics cannot select an interior mixture: λis therefore a learning hyperparameter. All the analytical findings are successfully checked against extensive Monte Carlo simulations.

cond-mat.dis-nn↗

arXiv2026

Partial annealing and pattern decorrelation in associative neural networks

Using the Hopfield model as a benchmark case, the present work focuses on the investigation of partially annealed associative neural networks, wherein neural dynamics is coupled to slowly evolving patterns within the two-temperature-two-timescale framework. This setting inherently introduces a real parameter n, reminiscent of the number of replicas in the celebrated replica trick, that tunes the separation of timescales and the effective interaction between fast (i.e. the neurons) and slow (i.e. the synapses) degrees of freedom. By adapting Guerra's interpolation to the case, we derive the free energy without relying on analytical continuation. The obtained results demonstrate that negative values of n induce a progressive decorrelation of the stored patterns, thereby effectively reducing interference, promoting orthogonal configurations and ultimately conferring to the network the maximal storage alphac=1. Numerical simulations based on a mean field Monte Carlo dynamics have been employed to confirm this scenario and prove that partial annealing restores retrieval in challenging regimes, such as in the presence of biased patterns, outperforming standard decorrelation methods. These findings underscore the notion of partial annealing as an adaptive mechanism for enhancing memory organisation and retrieval in complex systems.

cond-mat.dis-nn↗

arXiv2026

Dense Associative Memory with biased patterns: a Replica Symmetric analysis

We investigate dense higher-order associative memories in the high storage regime when the stored patterns are biased, namely when the entries of the patterns are not symmetrically distributed around zero. In this setting, the standard Hebbian prescription must be modified by recentering and rescaling the pattern entries, and an additional term must be introduced in the Hamiltonian to enforce consistency between the average activity of the network and that of the stored patterns. As a first step, we perform a signal-to-noise analysis in the zero-temperature limit and show that the bias reduces the effective storage capacity through a multiplicative correction factor (1-b^2)^P, while preserving the superlinear scaling with the system size. We then derive the quenched statistical pressure within the Replica Symmetric framework by means of Guerra's interpolation method and obtain the corresponding self consistency equations for the relevant order parameters. The analytical treatment confirms the heuristic prediction of the signal-to-noise argument, showing that the same bias dependent renormalization naturally emerges in the variance of the cross-talk noise. Finally, we discuss the resulting phase behavior of the model and its implications for retrieval performance in the model.

cond-mat.dis-nn↗

arXiv2026

Serial vs parallel recall in the Blume-Every-Griffiths neural networks

Fully connected Blume-Emery-Griffiths neural networks performing pattern recognition and associative memory have been heuristically studied in the past (mainly via the replica trick and under the replica symmetric assumption) as generalization of the standard Hopfield reference. In these notes, at first, by relying upon Guerra interpolation, we re-obtain the existing picture rigorously. Next we show that, due to dilution in the patterns, these networks are able to switch from serial recall (where one pattern is retrieved per time) to parallel recall (where several patterns are retrieved at once) and the larger the dilution, the stronger this emerging multi-tasking capability. In particular, we inspect the regimes of mild dilution (where solely a low storage of pattern can be enabled) and extreme dilution (where a medium storage of patterns can be sustained) separately as they give rise to different outcomes: the former displays hierarchical recall (distributing the amplitudes of the retrieved signals with different amplitudes), the latter -- instead -- performs a equal-strength recall (where a O(1) fraction of all the patterns is simultaneously retrieved with the same amplitude per pattern). Finally, in order to implement graded responses in the neurons, variations on theme obtained by enlarging the possible values of neural activity these neurons may sustain are also discussed generalizing the Ghatak-Sherrington model for inverse freezing in Hebbian terms.

cond-mat.dis-nn↗

arXiv2025

Yet another exponential Hopfield model

We propose and analyze a new variation of the so-called {\em exponential Hopfield model}, a recently introduced family of associative neural networks with unprecedented storage capacity. Our construction is based on a cost function defined through exponentials of standard quadratic loss functions, which naturally favors configurations corresponding to perfect recall. Despite not being a mean-field system, the model admits a tractable mathematical analysis of its dynamics and retrieval properties that agree with those for the original exponential model introduced by Ramsauer and coworkers. By means of a signal-to-noise approach, we demonstrate that stored patterns remain stable fixed points of the zero-temperature dynamics up to an exponentially large number of patterns in the system size. We further quantify the basins of attraction of the retrieved memories, showing that while enlarging their radius reduces the overall load, the storage capacity nonetheless retains its exponential scaling. An independent derivation within the perfect recall regime confirms these results and provides an estimate of the relevant prefactors. Our findings thus complement and extend previous studies on exponential Hopfield networks, establishing that even under robustness constraints these models preserve their exceptional storage capabilities. Beyond their theoretical interest, such networks point towards principled mechanisms for massively scalable associative memory, with potential implications for both neuroscience-inspired computation and high-capacity machine learning architectures.

cond-mat.dis-nn↗

arXiv2025

Guerra interpolation for inverse freezing

In these short notes, we adapt and systematically apply Guerra's interpolation techniques on a class of disordered mean-field spin glasses equipped with crystal fields and multi-value spin variables. These models undergo the phenomenon of inverse melting or inverse freezing. In particular, we focus on the Ghatak-Sherrington model, its extension provided by Katayama and Horiguchi, and the disordered Blume-Emery-Griffiths-Capel model in the mean-field limit, deepened by Crisanti and Leuzzi and by Schupper and Shnerb. Once shown how all these models can be retrieved as particular limits of a unique broader Hamiltonian, we study their free energies. We provide explicit expressions of their annealed and quenched expectations, inspecting the cases of replica symmetry and (first-step) broken replica symmetry. We recover several results previously obtained via heuristic approaches (mainly the replica trick) to prove full agreement with the existing literature. As a sideline, we also inspect the onset of replica symmetry breaking by providing analytically the expression of the de Almeida-Thouless instability line for a replica symmetric description: in this general setting, the latter is new also from a physical viewpoint.

cond-mat.dis-nn↗

arXiv2024

Boolean SK Model

For over half a century, statistical mechanics of spin glasses played as a paradigm to model and interpret disparate phenomena, ranging from quantitative biology to computer science. However, despite the extensive body of research in this area, there is still a notable lack of studies addressing the replacement of Ising spins with Boolean spins: as the latter play as bits in Machine Learning, this gap to fill is now mandatory. Purpose of this paper is to address this study by focusing on the mean field assumption, providing a comprehensive description of the results pertaining to these networks, referred to as the Boolean SK model due to their close relationship with the SK one. We provide a comprehensive framework for this model by employing Guerra interpolation: the thermodynamic limit, the replica symmetric and the broken replica free energy expressions are derived. Further, we inspect the onset of the replica symmetry breaking -- i.e., the de Almeida-Thouless line -- and derive Ghirlanda-Guerra fluctuations. All theoretical findings are corroborated by numerical inspections and both highlight crucial differences in the network's behavior if compared with the Ising SK model: as the temperature is lowered, no phase transitions are evidenced and the model continuously moves from a random (ergodic) behavior to a disordered (glassy) phase.

cond-mat.dis-nn↗

arXiv2024

Hebbian Learning from First Principles

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 relying upon maximum entropy extremization à la Jaynes. Beyond providing a formal derivation of these recipes for Hebbian learning, this construction also highlights how Lagrangian constraints within entropy extremization force network's outcomes on neural correlations: these try to mimic the empirical counterparts hidden in the datasets provided to the network for its training and, the denser the network, the longer the correlations that it is able to capture. Next, we prove that, in the big data limit, whatever the presence of a teacher (or its lacking), not only these Hebbian learning rules converge to the original storage prescription of the Hopfield model but also their related free energies (and, thus, the statistical mechanical picture provided by Amit, Gutfreund and Sompolinsky is fully recovered). As a sideline, we show mathematical equivalence among standard Cost functions (Hamiltonian), preferred in Statistical Mechanical jargon, and quadratic Loss Functions, preferred in Machine Learning terminology. Remarks on the exponential Hopfield model (as the limit of dense networks with diverging density) and semi-supervised protocols are also provided.

cond-mat.dis-nn↗

arXiv2024

About the AT line in Replica Symmetry Breaking assumption for spin glasses

Replica Symmetry Breaking is a fascinating phenomenon of spin glasses model which could have consequences also in other field of studies. Although there are several studies regarding the stability between the Replica Symmetric and first step of Replica Symmetry Breaking approximations, we have very few results for the following steps (apart from that one by Gardner for P-spin glasses in 1985 and Chen in 2017 and 2021). This is link to the fact that the classic method, based from the work by De Almeida and Thoules (from which the critical stability line takes its name), is cumbersome to generalise for the next assumptions. In this paper we devise a new straightforward method inspired to the work by Toninelli in 2002 to recover the critical line in order to inspect the stability first between the second and the first steps of Replica Symmetry Breaking and then, we generalise to Kth step, with K finite.

cond-mat.dis-nn↗

arXiv2023

Unsupervised and Supervised learning by Dense Associative Memory under replica symmetry breaking

Statistical mechanics of spin glasses is one of the main strands toward a comprehension of information processing by neural networks and learning machines. Tackling this approach, at the fairly standard replica symmetric level of description, recently Hebbian attractor networks with multi-node interactions (often called Dense Associative Memories) have been shown to outperform their classical pairwise counterparts in a number of tasks, from their robustness against adversarial attacks and their capability to work with prohibitively weak signals to their supra-linear storage capacities. Focusing on mathematical techniques more than computational aspects, in this paper we relax the replica symmetric assumption and we derive the one-step broken-replica-symmetry picture of supervised and unsupervised learning protocols for these Dense Associative Memories: a phase diagram in the space of the control parameters is achieved, independently, both via the Parisi's hierarchy within then replica trick as well as via the Guerra's telescope within the broken-replica interpolation. Further, an explicit analytical investigation is provided to deepen both the big-data and ground state limits of these networks as well as a proof that replica symmetry breaking does not alter the thresholds for learning and slightly increases the maximal storage capacity. Finally the De Almeida and Thouless line, depicting the onset of instability of a replica symmetric description, is also analytically derived highlighting how, crossed this boundary, the broken replica description should be preferred.

cond-mat.dis-nn↗

arXiv2023

About the de Almeida-Thouless line in neural networks

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 de Almeida and Thouless for spin glasses. The method is based on an expansion of the free-energy obtained within one-step of replica symmetry breaking (RSB) around the RS value. As such, it requires solely continuity and differentiability of the free-energy and it is robust to be applied broadly to systems with quenched disorder. We apply the method to the Hopfield model and to neural networks with multi-node Hebbian interactions, as case studies. In the appendices we test the method on the Sherrington-Kirkpatrick and the Ising P-spin models, recovering the AT lines known in the literature for these models, as a special limit, which corresponds to assuming that the transition from the RS to the RSB phase can be obtained by varying continuously the order parameters. Our method provides a generalization of the AT approach, which does not rely on this limit and can be applied to systems with discontinuous phase transitions, as we show explicitly for the spherical P-spin model, recovering the known RS instability line.

cond-mat.dis-nn↗

arXiv2023

Dense Hebbian neural networks: a replica symmetric picture of unsupervised learning

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 quality and quantity of the training dataset and the network storage, valid in the limit of large network size and structureless datasets. Moreover, we establish a bridge between macroscopic observables standardly used in statistical mechanics and loss functions typically used in the machine learning. As technical remarks, from the analytic side, we implement large deviations and stability analysis within Guerra's interpolation to tackle the not-Gaussian distributions involved in the post-synaptic potentials while, from the computational counterpart, we insert Plefka approximation in the Monte Carlo scheme, to speed up the evaluation of the synaptic tensors, overall obtaining a novel and broad approach to investigate neural networks in general.

cond-mat.dis-nn↗

arXiv2023

Dense Hebbian neural networks: a replica symmetric picture of supervised learning

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 parameters such as quality and quantity of the training dataset, network storage and noise, that is valid in the limit of large network size and structureless datasets: these networks may work in a ultra-storage regime (where they can handle a huge amount of patterns, if compared with shallow neural networks) or in a ultra-detection regime (where they can perform pattern recognition at prohibitive signal-to-noise ratios, if compared with shallow neural networks). Guided by the random theory as a reference framework, we also test numerically learning, storing and retrieval capabilities shown by these networks on structured datasets as MNist and Fashion MNist. As technical remarks, from the analytic side, we implement large deviations and stability analysis within Guerra's interpolation to tackle the not-Gaussian distributions involved in the post-synaptic potentials while, from the computational counterpart, we insert Plefka approximation in the Monte Carlo scheme, to speed up the evaluation of the synaptic tensors, overall obtaining a novel and broad approach to investigate supervised learning in neural networks, beyond the shallow limit, in general.

cond-mat.dis-nn↗

arXiv2021

Replica symmetry breaking in dense neural networks

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 statistical mechanical picture of the phenomenon of {\em replica symmetry breaking} (RSB) in these networks, then -- deepening results stemmed via these routes -- we aim to inspect the {\em glassiness} that they hide. In particular, regarding the methodology, we provide two techniques: the former is an adaptation of the transport PDE to the case, while the latter is an extension of Guerra's interpolation breakthrough. Beyond coherence among the results, either in replica symmetric and in the one-step replica symmetry breaking level of description, we prove the Gardner's picture and we identify the maximal storage capacity by a ground-state analysis in the Baldi-Venkatesh high-storage regime. In the second part of the paper we investigate the glassy structure of these networks: in contrast with the replica symmetric scenario (RS), RSB actually stabilizes the spin-glass phase. We report huge differences w.r.t. the standard pairwise Hopfield limit: in particular, it is known that it is possible to express the free energy of the Hopfield neural network as a linear combination of the free energies of an hard spin glass (i.e. the Sherrington-Kirkpatrick model) and a soft spin glass (the Gaussian or "spherical" model). This is no longer true when interactions are more than pairwise (whatever the level of description, RS or RSB): for dense networks solely the free energy of the hard spin glass survives, proving a huge diversity in the underlying glassiness of associative neural networks.

cond-mat.dis-nn↗

arXiv2021

Rigorous approaches for spin glass and Gaussian spin glass with P-wise interactions

Purpose of this paper is to face up to P-spin glass and Gaussian P-spin model, i.e. spin glasses with polynomial interactions of degree P > 2. We consider the replica symmetry and first step of replica simmetry breaking assumptions and we solve the models via transport equation and Guerra's interpolating technique, showing that we reach the same results. \\ Thus, using rigorous approaches, we recover the same expression for quenched statistical pressure and self-consistency equation in both assumption found with other techniques, including the well-known \textit{replica trick} technique. \\ At the end, we show that for $P=2$ the Gaussian P-spin glass model is intrinsecally RS.

cond-mat.dis-nn↗

arXiv2021

Pattern recognition in Deep Boltzmann machines

We consider a multi-layer Sherrington-Kirkpatrick spin-glass as a model for deep restricted Boltzmann machines and we solve for its quenched free energy, in the thermodynamic limit and allowing for a first step of replica symmetry breaking. This result is accomplished rigorously exploiting interpolating techniques and recovering the expression already known for the replica-symmetry case. Further, we drop the restriction constraint by introducing intra-layer connections among spins and we show that the resulting system can be mapped into a modular Hopfield network, which is also addressed rigorously via interpolating techniques up to the first step of replica symmetry breaking.

cond-mat.dis-nn↗

arXiv2020

Replica symmetry breaking in neural networks: a few steps toward rigorous results

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 drawn according to Hebb's learning rule): this is accomplished by grafting Guerra's telescopic averages on the transport equation technique, recently developed by some of the Authors. As an overture, we apply the technique to solve the Sherrington-Kirkpatrick model with i.i.d. Gaussian couplings centered at $J_0$ and with finite variance $J$; the mean $J_0$ plays the role of a signal to be detected in a noisy environment tuned by $J$, hence making this model a natural test-case to be investigated before addressing the Hopfield model. For both the models, an explicit expression of their quenched free energy in terms of their natural order parameters is obtained at the K-th step (K arbitrary, but finite) of replica-symmetry-breaking. In particular, for the Hopfield model, by assuming that the overlaps respect Parisi's decomposition (following the ziqqurat ansatz) and that the Mattis magnetization is self-averaging, we recover previous results obtained via replica-trick by Amit, Crisanti and Gutfreund (1RSB) and by Steffan and Kühn (2RSB).

math-ph↗