arXiv subjects

Francesco D'Amico

Publications and source records attributed to Francesco D'Amico.

7 recordsLinked to original sources

arXiv2026

Spectral phase transitions and trainability in neural network learning dynamics

The emergence of low-dimensional structures in the spectra of neural network weight matrices is a common empirical feature of trained models, but the dynamical origin of this phenomenon during learning remains an open problem. We formulate neural network training as the stochastic evolution of an initially random matrix ensemble, driven by stochastic gradient descent (SGD) updates that reshape the spectral bulk while amplifying signal strength. This induces a Baik-Ben Arous-Péché (BBP) transition during training, where isolated eigenvalues detach from the random bulk distribution, providing a dynamical framework for representation formation in high-dimensional learning dynamics. We demonstrate this in a solvable linear teacher-student model, where spectral evolution is analytically tractable and a phase diagram of trainability governed by the step size (or learning rate) and initial weight variance is obtained, and subsequently extend our formalism beyond the linear regime to nonlinear and stochastic settings. Numerical simulations in realistic settings support this picture, showing robust emergence of spectral alignment during training. Our results suggest that spectral analysis may provide a unified perspective of stochastic learning dynamics, linking trainability, optimisation hyperparameters, spectral phase transitions, and representation learning in neural networks.

cond-mat.dis-nn↗

arXiv2026

Implicit bias produces neural scaling laws in learning curves, from perceptrons to deep networks

Scaling laws in deep learning -- empirical power-law relationships linking model performance to resource growth -- have emerged as simple yet striking regularities across architectures, datasets, and tasks. These laws are particularly impactful in guiding the design of state-of-the-art models, since they quantify the benefits of increasing data or model size, and hint at the foundations of interpretability in machine learning. However, most studies focus on asymptotic behavior at the end of training. In this work, we describe a richer picture by analyzing the entire training dynamics: we identify two novel \textit{dynamical} scaling laws that govern how performance evolves as function of different norm-based complexity measures. Combined, our new laws recover the well-known scaling for test error at convergence. Our findings are consistent across CNNs, ResNets, and Vision Transformers trained on MNIST, CIFAR-10 and CIFAR-100. Furthermore, we provide analytical support using a single-layer perceptron trained with logistic loss, where we derive the new dynamical scaling laws, and we explain them through the implicit bias induced by gradient-based training.

cs.LG↗

arXiv2026

Benchmarking Graph Neural Networks in Solving Hard Constraint Satisfaction Problems

Graph neural networks (GNNs) are increasingly applied to hard optimization problems, often claiming superiority over classical heuristics. However, such claims risk being unsolid due to a lack of standard benchmarks on truly hard instances. From a statistical physics perspective, we propose new hard benchmarks based on random problems. We provide these benchmarks, along with performance results from both classical heuristics and GNNs. Our fair comparison shows that classical algorithms still outperform GNNs. We discuss the challenges for neural networks in this domain. Future claims of superiority can be made more robust using our benchmarks, available at https://github.com/ArtLabBocconi/RandCSPBench.

cond-mat.dis-nn↗

arXiv2025

Statistical mechanics of vector Hopfield network near and above saturation

We study analytically and numerically a Hopfield fully-connected network with $d$-dimensional vector spins. These networks are models of associative memory that generalize the standard Hopfield with Ising spins, where $P$ examples are stored in a network of $N$ units as local minima in an energy landscape. We study the equilibrium and out-of-equilibrium properties of the system, considering the system in its retrieval phase $α<α_c$ and beyond, where $α=P/N$ is the capacity of the system and $α_c$ is its critical value, above which storage fails. We derive the Replica Symmetric solution for the equilibrium thermodynamics of the system, together with its phase diagram: we find that the retrieval phase of the network shrinks with growing spin dimension, having ultimately a vanishing critical capacity $α_c\propto 1/d$ in the large $d$ limit. As a trade-off, we observe that in the same limit vector Hopfield networks are able to denoise corrupted input patterns in the first step of retrieval dynamics, up to very large capacities $α\propto d$. We also study the static properties of the system at zero temperature, considering the statistical properties of soft modes of the energy Hessian spectrum. We find that local minima of the energy landscape related to memory states have ungapped spectra with rare soft eigenmodes: these excitations are localized, their measure condensating on the noisiest neurons of the memory state.

cond-mat.dis-nn↗

arXiv2025

A Machine Learning Tool to Analyse Spectroscopic Changes in High-Dimensional Data

When nanoparticles (NPs) are introduced into a biological solution, layers of biomolecules form on their surface, creating a corona. Understanding how the structure of the protein evolves into the corona is essential for evaluating the safety and toxicity of nanotechnology. However, the influence of NP properties on protein conformation is not well understood. In this study, we propose a new method that addresses this issue by analyzing multi-component spectral data using Machine Learning (ML). We apply the method to fibrinogen, a crucial protein in human blood plasma, at physiological concentrations while interacting with hydrophobic carbon or hydrophilic silicon dioxide NPs, revealing striking differences in the temperature dependence of the protein structure between the two cases. Our unsupervised ML method a) does not suffer from the challenges associated with the curse of dimensionality, and b) simultaneously handles spectral data from various sources. The method offers a quantitative analysis of protein structural changes upon adsorption and enhances the understanding of the correlation between protein structure and NP interactions, which could support the development of nanomedical tools to treat various conditions.

cond-mat.stat-mech↗

arXiv2025

Pseudo-likelihood produces associative memories able to generalize, even for asymmetric couplings

Energy-based probabilistic models learned by maximizing the likelihood of the data are limited by the intractability of the partition function. A widely used workaround is to maximize the pseudo-likelihood, which replaces the global normalization with tractable local normalizations. Here we show that, in the zero-temperature limit, a network trained to maximize pseudo-likelihood naturally implements an associative memory: if the training set is small, patterns become fixed-point attractors whose basins of attraction exceed those of any classical Hopfield rule. We explain quantitatively this effect on uncorrelated random patterns. Moreover, we show that, for different structured datasets coming from computer science (random feature model, MNIST), physics (spin glasses) and biology (proteins), as the number of training examples increases the learned network goes beyond memorization, developing meaningful attractors with non-trivial correlations with test examples, thus showing the ability to generalize. Our results therefore reveal pseudo-likelihood works both as an efficient inference tool and as a principled mechanism for memory and generalization.

cond-mat.stat-mech↗

arXiv2024

Self-attention as an attractor network: transient memories without backpropagation

Transformers are one of the most successful architectures of modern neural networks. At their core there is the so-called attention mechanism, which recently interested the physics community as it can be written as the derivative of an energy function in certain cases: while it is possible to write the cross-attention layer as a modern Hopfield network, the same is not possible for the self-attention, which is used in the GPT architectures and other autoregressive models. In this work we show that it is possible to obtain the self-attention layer as the derivative of local energy terms, which resemble a pseudo-likelihood. We leverage the analogy with pseudo-likelihood to design a recurrent model that can be trained without backpropagation: the dynamics shows transient states that are strongly correlated with both train and test examples. Overall we present a novel framework to interpret self-attention as an attractor network, potentially paving the way for new theoretical approaches inspired from physics to understand transformers.

cs.LG↗