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Deep Patterns
The aim of this workshop is to bring together two scientific communities operating at the intersection of Applied Mathematics and Theoretical Physics. The former consists of researchers focused on “pattern formation” in several contexts (e.g., cellular automata, Turing patterns, etc.), while the latter includes those working on “pattern recognition” (e.g., neural networks, optimal estimation techniques, etc.). To date, these two groups have explored different but complementary facets of the concept of “pattern” that is becoming increasingly central across a wide spectrum of applied sciences (from detecting patterns in biomedical imaging for health screening, to understanding the genesis of recurrent structures in large language models or the study of morphologies in biological materials). With machine learning algorithms now able to detect patterns that often go beyond the reach of human perception — even when supported by traditional computing — the moment is right for a more collaborative approach. This workshop aims to encourage the formation of a shared community, where diverse methods and viewpoints come together to explore the complex nature of patterns in order to support scientific progress in an area that is increasingly central to modern applied science.
Organizers
- Wioletta Ruszel, Utrecht University
- Cristian Spitoni, Utrecht University
- Emilio Cirillo, Sapienza Università di Roma
- Adriano Barra, Sapienza Università di Roma
Speakers and program
The program starts on Monday 18 May at 9.30 and will end on Friday 22 May at 15.00h. You can download the detailed program here.
Speakers are:
| Elena Agliari | Sapienza University of Rome | abstract |
| Linda Albanese | Università del Salento | abstract |
| Luca Ambrogioni | Radboud University | abstract |
| Robbin Bastiaansen | Utrecht University | abstract |
| Daniele Andreucci | Sapienza Università di Roma | abstract |
| Christian Bick | Free University Amsterdam | abstract |
| Fabio Coppini | Utrecht University | abstract |
| Anthony CC Coolen | Radboud University | abstract |
| Alessandro Corbetta | Eindhoven University of Technology | abstract |
| Paolo dai Pra | Università di Verona | abstract |
| Nazim Fates | Inria Nancy, University of Lorraine | abstract |
| Massimo Frittelli | University of Salento | abstract |
| Federica Gerace | Università di Bologna | abstract |
| Alessandro Ingrosso | Radboud University | abstract |
| Vanesssa Jacquier | University of Padova | abstract |
| Maike de Jongh | Twente University | abstract |
| Sander Hille | Leiden University | abstract |
| Tim Kamsma | Utrecht University | abstract |
| Andrea Ladiana | Sapienza Università di Roma | abstract |
| Andrea Lepre | Sapienza Università di Roma | abstract |
| Carlo Lucibello | Bocconi University | abstract |
| Emanuele Massa | Radboud UMC | abstract |
| Adrian Muntean | Karlstad University | abstract |
| Mark Peletier | Eindhoven University of Technology | abstract |
| Lars Sickert Karam | Eindhoven University of Technology | abstract |
| Daniele Tantari | University of Bologna | abstract |
Registration
Registration has been closed!
If you want to know if there is still a possibility to participate, please send an email to eurandom.office@tue.nl .
Abstracts
Elena Agliari
Hebbian Learning and the Art of Pattern Maintenance
Hebb’s rule provides a prototypical mechanism for storing information in neural networks through simple, local interactions among constituent neurons. In this talk, we examine Hebbian networks from a statistical-mechanical perspective, focusing on the conditions under which stored memories, also referred to as patterns, can be reliably retrieved. Specifically, neurons are modeled as binary units that evolve according to Gibbs dynamics, where relaxation to a stable configuration is interpreted as the retrieval of the patterns encoded in that configuration. Emphasis is placed on the influence of network architecture and scaling regimes on memory capacity and robustness: This perspective highlights how pattern retrieval is shaped by the structure of the energy landscape induced by Hebbian interactions, and allows us to design optimal variants of the original rule including scenarios where the stored patterns are noisy or incomplete.
Linda Albanese
The emerging of Inverse Freezing in neural networks: parallel and serial processing
In this talk we present a rigorous statistical mechanical analysis of Blume-EmeryGriffiths (BEG) and Ghatak-Sherrington (GS) neural networks, two multi state generalisations of the Hopfield model. Using Guerra’s interpolation method, we derive the replica symmetric free energies and the corresponding self consistency equations governing pattern retrieval without relying on heuristic replica techniques. We then study how pattern dilution affects retrieval in low and medium storage regimes. Mild dilution induces a hierarchical retrieval scenario, where stored patterns emerge with different amplitudes. By contrast, strong dilution produces a multitasking phase in which several patterns can be retrieved simultaneously with comparable strength. Finally, we discuss how graded neuronal responses in GS networks modify these parallel retrieval regimes and their computational implications.
This talk is inspired by the joint work with Andrea Alessandrelli (University of Salento) and Adriano Barra and Emilio N.M. Cirillo (Sapienza University of Rome)
Luca Ambrogioni
Symmetry breaking, entropy production and pattern formation in generative diffusion
Generative diffusion models can be understood as stochastic physical systems driven far from equilibrium, where structure emerges through controlled noise dissipation. In this talk, I present a unified perspective that connects diffusion-based generation to concepts from statistical physics and information theory, focusing on three intertwined phenomena: symmetry breaking, entropy production, and pattern formation. I will argue that sampling in diffusion models proceeds through distinct dynamical phases. Early dynamics remain close to a high-symmetry, noise-dominated fixed point, while later stages undergo symmetry-breaking instabilities that steer trajectories toward low-dimensional data manifolds. These instabilities act as critical points where trajectories branch, diversity is created, and semantic structure first appears. From an information-theoretic viewpoint, this process is accompanied by nontrivial entropy production: the score field selectively suppresses incompatible noise modes while amplifying informative directions, effectively acting as a nonlinear filter that injects information into the generative process. By linking spectral properties of the score Jacobian, entropy production rates, and dynamical phase transitions, the talk provides a coherent account of how diffusion models form patterns, balance diversity and fidelity, and remain robust despite operating in extremely high-dimensional spaces. This framework suggests that high-quality generation is not achieved by avoiding instability, but by shaping and exploiting it.
Daniele Andreucci
Fick’s vs Fokker-Planck: diffusion in heterogeneous media, a homogenization approach.
The two parabolic partial differential equations of Fick and Fokker-Planck have both been proposed to model diffusion in the presence of inhomogeneities. Here we understand inhomogeneities as inclusions where diffusivity is vanishingly small. We present some results of our investigation of upscaled models obtained via the mathematical theory of homogenization, attempting to track specifically the asymptotic behavior of the distribution of mass, besides determining the equations satisfied by the limiting solution. This requires a careful analysis of the scalings obeyed by the inclusions’ size and by the degeneration of diffusivity inside them.
Robbin Bastiaansen
The dynamics of spatial patterns in nature
Spatial patterns arise across a remarkable range of natural systems – such as vegetation patterns in drylands, patterned stratocumulus clouds, convection cells in ocean and atmosphere, clustering in animal populations, spatial interfaces between tropical rainforests and savanna ecosystems, melt ponds in ice sheets, and mussel bed formation. In this talk, I will give an overview, based on theory of reaction-diffusion equations, of how such spatial patterns might emerge and how they evolve under changing conditions. In particular, I will talk about tipping points/bifurcations and transitions to and from spatial patterned systems – as in the natural systems these could indicate e.g. irreversible loss of ecosystem functioning or irreversible changes to the climate system. During the talk, I will highlight research gaps in the theory, generalizability of the theory, and the real systems.
Christian Bick
Synchrony patterns in finite oscillator networks and their continuum limits
ynchrony in networks of coupled oscillatory units are collective dynamics characterized by the dynamical nodes behaving in unison. Apart from global synchrony that involves all units across the network, synchrony patterns – different parts of a networks show different synchrony properties – have attracted attention. First, we discuss how synchrony patterns arise in coupled oscillator networks and how they link to network symmetries. Second, we consider the symmetries of continuum limits of network dynamics. That helps identify different synchrony patterns in the infinite-dimensional dynamics of the limit.
Anthony CC Coolen
Extracting patterns from data using statistical physics
In this talk I discuss a variety of topics at the interface between data analytics and statistical physics that I find interesting or potentially under-explored. The first half is about applications and deals with regression problems in modern medicine (including responder identification, competing risks, Bayesian federated inference, and overfitting correction). The second half is about mathematical methodology, and looks at navigating complex parameter spaces, the power of the replica method, and quantum computing challenges.
Fabio Coppini
Language agnostic key-value extraction using logical neural networks
Humans exhibit the ability to interpret document structure—such as forms, tables, and key-value relations—largely independently of the underlying language. In this work, we use Logical Neural Networks (Riegel et al, 2020) to investigate whether logical patterns arise in the context of key-value extraction task. Our results show that a predicate space derived from layout only predicates is enough to learn rules to predict key-value pairs in a given document page. Moreover, we are also able to show that these patterns are somehow invariant with respect to the language, i.e., different languages produce similar (if not equal) logical rules.
Alessandro Corbetta
Modeling randomness in pedestrian routing
Understanding and quantitatively modeling pedestrian dynamics remains a key challenge with broad implications for the safety, efficiency, and serviceability of public spaces. At the same time, it offers a fascinating window into the statistical physics of active matter.
This talk explores the stochastic nature of pedestrian routing, addressing both individual and collective (N-body) behaviors. Using multi-million–trajectory datasets from real-world observations, we examine the emergence of variability and randomness in pedestrian decisions and paths.
We focus on the interplay between, and identification of, variational principles and noisy effects. Assuming pedestrians pursue an approximate optimality – balancing travel time and perceived comfort – we discuss how stochasticity, arising from inter-individual differences, limited information, and environmental uncertainty, leads to systematic deviations from optimal motion. Both discrete and continuous modeling approaches are discussed, highlighting identification challenges and current level of (statistically) quantitative predictions.
Paulo Dai Pra
Critically and nonlinearity ina model of price formation
Many natural and social phenomena exhibit patterns such as heavy tails, multifractality and nonlinearity. In some cases these phenomena are well modeled by complex systems, inspired by statistical physics, at the critical point. We present here a stochastic multi-agent model, for price formation in a simple market. After having identified a critical point, we show that the macroscopic evolution of the price follows a stochastic volatility model with nonlinear mean reversion. These nonlinear models have been recently introduced in the financial/econometric literature to model anomalous patterns in price dynamics. The aim of this research is to provide a microscopic foundation for these models.
Joint work with Paolo Pigato, Università di Roma II.
Nazim Fates
Phase transitions in Probablistic Cellular Automata
Since the seminal work of Turing in 1952, it has been understood that cellular automata, when combined with randomness, can generate a rich variety of patterns and implement diverse symmetry-breaking mechanisms. Probabilistic cellular automata (PCA), in particular, are known to exhibit various types of phase transitions. This raises a series of questions. ‘Where’ do such phenomena lie within the “natural” mathematical spaces that define the simplest PCA models? What produces these phase transitions and what kinds of patterns do they produce? To which extent can their behavior be predicted analytically? May we even ‘use’ them to solve computational problems? In this talk, I propose to investigate these questions through selected examples drawn from recent and less recent research.
Massimo Frittelli
Turing patterns: modeling issues, computational challenges and deep learning applications
Over the past decade, we have developed a reaction-diffusion PDE model able to capture the essential features of unstable material growth in electrodeposition and describe them in terms of Turing pattern formation. Recharge instability problems in batteries are a special case of this phenomenon. In presence of diffusion-driven or Turing instability, the numerical approximation of the RD-system (called DIB model) attains at the “steady state” solutions with different intriguing spatial morphologies, called Turing patterns, like spots, holes, stripes and labyrinths[1]. To speed up the computations, for a selection of numerical methods we present the matrix-oriented approach recently introduced in [2,3]. Moreover, we show some recent results about Turing pattern formation in 3D: on closed surfaces [4], in bulk-surface domains [5] and on evolving surfaces [6]. In the second part, we focus on comparisons with experiments and related model parameter identification, a crucial problem to quantitatively link the battery response to numerical simulations. As an alternative to the PDE constrained minimization [7], we propose the Deep-Learning approach in [8] applying a Convolutional Neural Network (CNN) trained on numerical solutions of the DIB model. We show: i) morphological classification of simulated and experimental patterns; ii) parameter identification for experimental images; iii) robustness with respect to electrode image rotations and scale-changes.
Joint work with: B. Bozzini (Polytechnic of Milan), M. Frittelli (UniSalento), D.Lacitignola
(Uni Cassino), A. Lawless (Univ Reasing UK), A. Madzvamuse (British Columbia Canada), V.
Simoncini (UniBologna)
Federica Gerace
Testing Transformer Learnability on the Arithmetic Sequence of Rooted Trees
We study whether a Large Language Model can learn the deterministic sequence of trees generated by the iterated prime factorization of the natural numbers. Each integer is mapped into a rooted planar tree and the resulting sequence defines an arithmetic text with measurable statistical structure. A transformer network (the GPT-2 architecture) is trained from scratch on the first elements to subsequently test its predictive ability under next-word and masked-word prediction tasks. Our results show that the model partially learns the internal grammar, capturing non-trivial regularities and correlations. This suggests that learnability may extend beyond empirical data to the very structure of arithmetic.
Vanessa Jacquier
From Nonlocal Perimeters to Critical Droplets in the Long-Range Bi-Axial Ising Model
We introduce a nonlocal generalization of the classical perimeter, the \emph{nonlocal bi-axial discrete perimeter}, in which long-range interactions contribute along the coordinate axes, so that not only the external boundary of a polyomino $P$, but all its internal and external components enter the energy.
For $\lambda>1$, the nonlocal perimeter is defined as\[ Per_{\lambda}(\mathcal{P}):=\sum_{x \in \mathbb{Z}^2 \cap \mathcal{P}, \, y \in \mathbb{Z}^2 \cap \mathcal{P}^c} \frac{1}{d^{\lambda}(x,y)},\] where the interaction decays as a power law and is restricted to axial directions. We analyze the associated nonlocal discrete isoperimetric problem, characterizing minimizers among polyominoes of fixed area.
In the context of metastability for the \emph{long-range bi-axial Ising model}, these minimizers acquire a precise dynamical meaning: they identify the shape and size of the critical droplet that the system must nucleate in order to transition from a metastable state to the stable phase. Their energy determines the height of the barrier between the metastable and stable states, and thus directly controls the transition time, which grows exponentially with the barrier under Glauber dynamics in the metastable regime. In this way, the isoperimetric problem provides a geometric characterization of the nucleation mechanism, linking optimal shapes, energy barriers, and transition times within a unified framework.
Maike de Jongh
Controlled stochastic growth of structures in the zero-temperature Ising model
The optimization of structure growth in spatially stochastic systems poses a compelling challenge in a wide range of applications, including biological pattern formation and solution-based crystallization. In this talk, we investigate control strategies for a two-dimensional zero-temperature Ising model on a finite square lattice evolving under Metropolis dynamics.
We consider a setting in which an external controller aims to drive the system from a configuration containing one or two small droplets of +-spins toward the all-plus configuration by flipping selected spins at prescribed times. To analyze this control problem, we formulate it as a Markov decision process (MDP), a classical framework for sequential decision-making problems under uncertainty.
To compute an optimal policy, we construct a simplified model by reducing the configuration space to the local minima of the Hamiltonian. Leveraging structural properties of this simplified model, we characterize the optimal policy by solving the Bellman equations in a recursive manner.
Finally, we present simulation results demonstrating the performance of the optimal policy and illustrating the induced growth mechanism. Moreover, we show that its geometric features persist at low but positive temperatures.
Sander Hille
Quantitative comparison of patterns: a role for measures?
UMeasures and metrics on the space of measures have been used in the field of image analysis. One application is to the denoising problem. A denoised image is then obtained by translating the image into a measure and find a measure that is minimizing a functional defined with a form of dual bounded Lipschitz norm. On the other hand, metrics from the Wasserstein family and their relation to optimal transport have been employed to determine the flow of ‘mass’ between images in a timelapse movie. Such applications will be reviewed, followed by discussion of another practical example: quantitative comparison of data of stomatal patterning on leaves with those of model simulations, needed for parameter estimation in the model. Among others, this question led to an approach to compute Fortet-Mourier distances between measures, which was not possible before. This approach will be highlighted.
Alessandro Ingrosso
Generalization performance of narrow one-hidden layer networks in the teacher-student setting
Understanding the generalization abilities of neural networks for simple input-output distributions is crucial to account for their learning performance on real datasets. The classical teacher-student setting, where a network is trained from data obtained thanks to a label-generating teacher model, serves as a perfect theoretical test bed. In this work, we give a complete theoretical account of the teacher-student generalization performance of fully connected one-hidden layer networks with generic activation functions, in a regime where the number of hidden units is large, yet much smaller than the input dimension. Using methods from statistical physics, we provide closed-form expressions for the typical performance of both finite temperature (Bayesian) and empirical risk minimization estimators. In doing so, we highlight the presence of a transition where hidden neurons specialize when the number of samples is sufficiently large and proportional to the number of parameters of the network. Our theory accurately predicts the generalization error of neural networks trained on regression or classification tasks with either noisy full-batch gradient descent (Langevin dynamics) or full-batch gradient descent.
Tim Kamsma
Iontronic Dynamical Systems for Neuromorphic Computing in a Brain-Inspired substrate
Neuromorphic computing aims to replicate the information processing of the human brain, which remains unparallelled in its capacity for general intelligence and energy efficiency. Artificial aqueous memristors that directly mimic the fluidic ion transport found in the brain have emerged as an exciting new platform for both brain-inspired signalling and computing. The involved physics behind these memristors can be reduced to the relevant underlying mathematical formalisms. This enables the characterisation of iontronic neuromorphic spiking circuits as mathematically treatable two-dimensional dynamical systems. Moreover, we can show that the governing physical equations of such memristors enable a circuit design that exhibits a one-to-one equivalence with the mathematical definitions of the established recurrent neural network implementations of echo state and band-pass networks. We successfully analyse chaotic Mackey-Glass time series and real-world respiratory pressure data, where notably the lung pressures can directly couple to the circuit’s internal dynamics via iontronic’s intrinsic pressure responsiveness.
Andrea Ladiana
Distributed Pattern Recognition via Federated Hopfield Networks
Federated learning enables collaborative training without sharing raw data, but struggles under client heterogeneity and streaming distribution shifts, where drift and novel data can impair convergence and cause forgetting. We propose a federated associative-memory framework that learns shared archetypes in heterogeneous, continual settings, where client data are independent but not necessarily balanced. Each client encodes its experience as a low-rank Hebbian operator, sent to a central server for aggregation and factorization into global archetypes. This approach preserves privacy, avoids centralized replay buffers, and is robust to small, noisy, or evolving datasets. We cast aggregation as a low-rank-plus-noise spectral inference problem, deriving theoretical thresholds for detectability and retrieval robustness. An entropy-based controller balances stability and plasticity in streaming regimes. Experiments with heterogeneous clients, drift, and novelty show improved global archetype reconstruction and associative retrieval, supporting the spectral view of federated consolidation.
Andrea Lepre
Emergent Cooperativeness in Three-Directional Associative Memories
In this talk, we introduce a unified learning framework for Three-Directional Associative Memory (TAM) models, extending the classical Hebbian paradigm to multi-layer architectures under both supervised and unsupervised protocols. By leveraging techniques from the statistical mechanics of disordered systems, specifically Guerra’s interpolation and replica theory, we provide an analytical description of the network’s storage capacity and computational phase transitions.
A key focus of this talk is the phenomenon of “cooperativeness” emerging among interconnected layers. Our analysis reveals that retrieval performance is not isolated to local layer noise; instead, layers trained on low-entropy datasets actively stabilize those exposed to higher noise, synergistically balancing retrieval boundaries across the network. Validated by Monte Carlo simulations, these findings offer a robust blueprint for designing resilient hetero-associative systems capable of pattern completion and disentanglement.
Carlo Lucibello
Memories and Diffusions: from dense associative memories to generative diffusion.
Recent generalizations of the Hopfield model of associative memory can store a number P of random patterns that grows exponentially with the number N of neurons. I will present a formalism, based on the Random Energy Model from spin glass theory, that allows for a precise characterization of the model’s maximum capacity and its basins of attraction. I will then show how this same formalism can be extended to analyze memorization phenomena in generative diffusion models. In particular, within a simplified setting, we show that the number of training points required to avoid adverse memorization effects is significantly reduced when the data manifold has a much lower dimensionality than the ambient space. Finally, I will present additional results, from a statistical physics perspective, on the key features of diffusion dynamics as they approach the data manifold.
Emanuele Massa
Statistical mechanics of Cox regression in the proportional regime
In this talk I will introduce the Cox regression for time to event data and explain how to develop its asymptotic theory in the proportional regime (where the number of model features scales proportionally with the number of samples).
The asymptotic theory is constructed by means of the Replica method, from the theory of spin glasses, under the (ideal) assumption of uncorrelated gaussian features and when the regularized Cox loss is strongly convex. The resulting theory depends on unknowns like the true signal strength and the true shape of the data generating process. Methodology is proposed to obtain finite sample size versions of the RS equations, and hence of the RS order parameter of the theory directly from the data.
Adrian Muntean
How Evaporation Drives Phase Separation in Ternary Mixtures: A Non-Local Continuum Approach
Motivated by experiments on thin films cast from ternary solutions containing two solutes and a volatile solvent, we investigate morphology formation in three-state systems using both a lattice (microscopic) model and its continuum (macroscopic) analogue. The lattice formulation is based on the Blume–Capel model with conservative Kawasaki dynamics, while the continuum description consists of coupled evolution equations with nonlinear nonlocal drift terms obtained as a hydrodynamic limit of a Kac-type interaction. We perform 2D and 3D simulations to examine how morphology depends on solvent content and how evaporation reshapes the geometry and connectivity of emerging domains. We also provide a well-posedness result for the continuum system and outline its proof, and we demonstrate that our finite-volume schemes accurately approximate the corresponding weak solution.
This talk reports on joint work with N. Jävergård and S. A. Muntean (Karlstad, SWE), R. Lyons (Boulder, USA), and E. N.M. Cirillo (Rome,IT), supported by the Swedish Research Council (nr. 2024-05606)
Mark Peletier
Noisy dislocations: 2D Coulomb interaction with Brownian noise
Some Keller-Segel PDEs can be viewed as the many-particle limit of a system of stochastic particles that attract each other. The particles experience Brownian noise, and in the critical case of Coulomb attraction the interplay between attraction and noise gives rise to highly non-trivial behavior, as was shown by Fournier, Jordain, and Tardy.
In this paper we change the microscopic particle system from all-pair attraction to a signed attraction-repulsion, in which opposite signs attract and equal signs repel each other. This is inspired both by colloids that can be electrically charged and by 2D dislocation models that have similar `charges’. In both cases the interaction again has Coulomb scaling and is therefore critical.
We show that for this `electronic’ attraction-repulsion setup the situation is different than for the all-attracting case, and we give a detailed description of collision and de-collision events. We show that for this type of interaction the many-particle limit is well posed for interaction strengths that grow faster with N than in the case of all-attracting particles. In particular, the many-particle case is well-posed in the trivial-scaling case of `zooming out’ while preserving total particle charge.
Finally, we establish a many-particle, mean-field limit to a Keller-Segel-type PDE.
This is joint work with Thomas Slangen (Eindhoven) and Patrick van Meurs (Kanazawa, Japan).
Lars Sickert Karam
Challenges in inference for inhomogeneous spatial point processes and applications to waiting crowds
Statistical inference for interacting spatial point processes in settings with inhomogeneous background intensities is fundamentally challenging, as a clean decoupling of interaction mechanisms from environmental influences (i.e., the background field) is typically infeasible. While progress has been made in strictly parametric settings, semiparametric or nonparametric approaches remain comparatively underexplored. This talk presents waiting crowds as a compelling case study, in which the unique availability of multiple independent samples from the same process enables the study of inference in a highly heterogeneous setting with complex interaction structures. We introduce semiparametric models for determinantal and Gibbs point processes suitable for this context, together with an inference methodology allowing a quantitative fitting of these models to data. We demonstrate these models are able to reproduce key empirical features of the process of waiting in pedestrians, despite persistent challenges in separating the influences of inhomogeneity from interaction. Beyond setting the scene for future research in spatial point processes, in particular inference with replicated spatial patterns, our findings are also societally relevant, as quantitative models for waiting pedestrians are crucial for designing transportation infrastructure and understanding agent-based physical systems.
Joint work with Rui M. Castro and Alessandro Corbetta
Daniele Tantari
Permutation Symmetry Breaking in Teacher–Student Restricted Boltzmann Machines
Restricted Boltzmann machines (RBM) are generative models capable to learn data with a rich underlying structure. We study the teacher–student setting where a student RBM learns structured data generated by a teacher RBM. When teacher patterns are uncorrelated the learning dynamics undergo a transition in which student hidden units align one‑to‑one with teacher patterns, independently of network width. When correlations are introduced, the symmetric phase splits into a sequence of Permutation symmetry breaking transitions, each reflecting a different degree of feature disentanglement. These transitions determine the critical data load for successful learning and reveal how RBMs internal representations specialize under structured data.