arXiv subjects

Fabio Anselmi

Publications and source records attributed to Fabio Anselmi.

At least 19 recordsLinked to original sources

arXiv2026

PAC--Bayes Bounds on Quotient Parameter Spaces: Geometry-induced Implicit-Bias Priors

Overparameterized models often have continuous parameter symmetries, so different parameters define the same predictor. We show that PAC--Bayesian analysis should be performed on the quotient predictor space: pushing a prior and posterior to the quotient preserves the empirical and population Gibbs risks while removing the nonnegative KL contribution caused solely by how the two distributions differ among parameterizations of the same predictor. Quotienting alone does not determine which prior to use. We construct a canonical choice of one parameterization for each predictor and account for the geometric volume of its equivalent parameterizations. This transforms a neutral reference prior into a data-independent prior that reflects the model's implicit bias. It approximates the ideal but inadmissible posterior-matched prior, which would minimize the KL term by depending on the training data. The resulting certificate is tighter exactly when this geometry-induced prior has smaller KL divergence from the learned quotient posterior than the neutral prior. We test this prediction in Fourier regression with a Hadamard parameterization and in Query-Key attention, using ordinary SGD without an explicit regularizer. The implicit-bias prior reduces the mean quotient-space KL by $40.69\%$ and the mean PAC--Bayes certificate by $21.40\%$ in the Fourier-Hadamard experiment. The smaller, prior-scale-dependent improvement in Query-Key attention confirms the predicted conditional nature of the effect.

cs.LG↗

arXiv2026

Attractor Geometry Determines the Identifiability Limits of System Discovery

Symbolic discovery of governing equations from data is limited not only by algorithm design and data volume, but by the geometry of the attractor: what the long-run dynamics allow to be recovered. Using a within-system design on Lorenz-84, where one forcing parameter drives fixed-point, limit-cycle, and chaotic regimes while the governing equations and library stay fixed, we show that a single number, $\lambda_{\min}(M)$, the smallest eigenvalue of the invariant-measure moment matrix, sets the identifiability ceiling for both sparse regression (SINDy) and evolutionary symbolic regression (PySR). Derived from the Birkhoff ergodic theorem and obtained from a short reference trajectory before any run, $\lambda_{\min}(M)$ measures how fully the attractor covers function space: where it vanishes, recovery is impossible for any algorithm, sparse or combinatorial alike; as it grows, both algorithms improve. Chaos raises $\lambda_{\min}(M)$ by spreading the attractor, but also enlarges it and amplifies noise; because noise enters SINDy's regression bottleneck linearly and PySR's discrimination channel superlinearly, the same transition can push the two methods in opposite directions, so deeper chaos is not uniformly better. Parameter-free mechanistic scores from this framework transfer without refitting to a held-out Lorenz-96 system, confirming mechanism rather than curve-fitting; a criterion read from the equations predicts when added chaos will not improve conditioning. We also introduce Soft F1, a coefficient-weighted structural metric that resolves performance differences invisible to binary-success and predictive scores. The first question of discovery is then not which algorithm, but what the attractor permits.

cs.LG↗

arXiv2026

Sparse positive maps on qutrits with exact nondecomposability thresholds and PPT-entanglement transitions

We study a family of sparse positive maps on qutrits for which positivity, decomposability, and PPT entanglement can all be analysed explicitly. The block structure of the associated Choi matrices reduces positivity to a Hermitian biquadratic form and leads to exact positivity boundaries for three representative parametric families. For the same families we determine the exact transition between decomposable and non-decomposable maps and construct associated PPT states of two classes. The first consists of witness-adapted deformations naturally tied to the non-decomposability analysis. The second consists of analytically tractable families whose full PPT-entangled branch is detected by fixed positive maps, yielding exact thresholds between separability and bound entanglement. For the trace-preserving subclass, we further compare positivity with a recent eigenvalue bound for 2-positive maps, thereby making the gap between positivity and higher-order positivity fully explicit within this family.

quant-ph↗

arXiv2026

Generating Non-Decomposable Maps with Differentiable Semidefinite Programming

Positive maps that are not decomposable are a key resource in entanglement theory because they can detect bound entangled states, yet systematic methods for constructing them remain limited. We introduce an optimization framework based on differentiable semidefinite programming (SDP) for generating positive non-decomposable maps under flexible structural constraints on their Choi matrices. The method combines SDP-based certificates of non-decomposability and positivity with gradient-based optimization, enabling a systematic search over maps with different input and output dimensions. Within this framework, we generate previously unknown numerical examples, identify a parametrized family of maps arising from masked Choi matrices, and construct real non-decomposable maps. We further show that the same approach can be adapted to explore open questions in quantum information theory, including the PPT square conjecture and recently proposed eigenvalue bounds for 2-positive trace-preserving maps.

quant-ph↗

arXiv2026

Perceptual misalignment of texture representations in convolutional neural networks

Mathematical modeling of visual textures traces back to Julesz's intuition that texture perception in humans is based on local correlations between image features. An influential approach for texture analysis and generation generalizes this notion to linear correlations between the nonlinear features computed by convolutional neural networks (CNNs), compiled into Gram matrices. Given that CNNs are often used as models for the visual system, it is natural to ask whether such "texture representations" spontaneously align with the textures' perceptual content, and in particular whether those CNNs that are regarded as better models for the visual system also possess more human-like texture representations. Here we quantify the perceptual content captured by feature correlations computed for a diverse pool of CNNs, and we compare it to the models' perceptual alignment with the mammalian visual system as measured by Brain-Score. Surprisingly, we find that there is no connection between conventional measures of CNN quality as a model of the visual system and its alignment with human texture perception. We conclude that texture perception involves mechanisms that are distinct from those that are commonly modeled using approaches based on CNNs trained on object recognition, possibly depending on the integration of contextual information.

cs.CV↗

arXiv2026

Zigzag Persistence of Neural Responses to Time-Varying Stimuli

We use topological data analysis to study neural population activity in the Sensorium 2023 dataset, which records responses from thousands of mouse visual cortex neurons to diverse video stimuli. For each video, we build frame-by-frame cubical complexes from neuronal activity and apply zigzag persistent homology to capture how topological structure evolves over time. These dynamics are summarized with persistence landscapes, providing a compact vectorized representation of temporal features. We focus on one-dimensional topological features-loops in the data-that reflect coordinated, cyclical patterns of neural co-activation. To test their informativeness, we compare repeated trials of different videos by clustering their resulting topological neural representations. Our results show that these topological descriptors reliably distinguish neural responses to distinct stimuli. This work highlights a connection between evolving neuronal activity and interpretable topological signatures, advancing the use of topological data analysis for uncovering neural coding in complex dynamical systems.

math.AT↗

arXiv2026

Understanding and inverse design of implicit bias in stochastic learning: a geometric perspective

A key challenge in machine learning is to explain how learning dynamics select among the many solutions that achieve identical loss values in overparameterized models - a phenomenon known as implicit bias. Controlling this bias provides a direct mechanism on learned representations, which are central to interpretability, robustness, and reasoning in modern AI systems. Yet, despite its importance, existing explanations remain largely ad hoc and lack a unifying mechanism. We develop a theoretical and constructive framework in which implicit bias emerges as a geometric correction induced by the interplay between gradient noise and continuous symmetries of the loss. We compute the induced bias across a range of architectures, predicting new behaviors and explaining known ones. The approach also enables inverse design: by engineering predictor - preserving parameterizations, it is possible to shape the bias, with sparsity and spectral sparsity emerging as canonical instances. Numerical experiments support the theory and validate the inverse - design framework in controlled settings.

cs.LG↗

arXiv2025

The Spectral Bias of Shallow Neural Network Learning is Shaped by the Choice of Non-linearity

Despite classical statistical theory predicting severe overfitting, modern massively overparameterized neural networks still generalize well. This unexpected property is attributed to the network's so-called implicit bias, which describes its propensity to converge to solutions that generalize effectively, among the many possible that correctly label the training data. The aim of our research is to explore this bias from a new perspective, focusing on how non-linear activation functions contribute to shaping it. First, we introduce a reparameterization which removes a continuous weight rescaling symmetry. Second, in the kernel regime, we leverage this reparameterization to generalize recent findings that relate shallow Neural Networks to the Radon transform, deriving an explicit formula for the implicit bias induced by a broad class of activation functions. Specifically, by utilizing the connection between the Radon transform and the Fourier transform, we interpret the kernel regime's inductive bias as minimizing a spectral seminorm that penalizes high-frequency components, in a manner dependent on the activation function. Finally, in the adaptive regime, we demonstrate the existence of local dynamical attractors that facilitate the formation of clusters of hyperplanes where the input to a neuron's activation function is zero, yielding alignment between many neurons' response functions. We confirm these theoretical results with simulations. All together, our work provides a deeper understanding of the mechanisms underlying the generalization capabilities of overparameterized neural networks and its relation with the implicit bias, offering potential pathways for designing more efficient and robust models.

cs.LG↗

arXiv2024

Decoherence Cancellation through Noise Interference

We propose a novel, feedback-free method to cancel the effects of decoherence in the dynamics of open quantum systems subject to dephasing. The protocol makes use of the coupling with an auxiliary system when they are both subject to the same noisy dynamics, in such a way that their interaction leads to cancellation of the noise on the system itself. This requires tuning the strength of the coupling between main and auxiliary systems as well as the ability to prepare the auxiliary system in a Fock state, which solely depends on the coupling strength. We investigate the protocol's efficiency to protect NOON states against dephasing in setups such as tweezers arrays of cold atoms. We show that the protocol's efficiency is robust against fluctuations of the optimal parameters and, remarkably, that it is independent of the temporal noise features. Therefore, it can be applied to cancel both Markovian and non-Markovian noise effects, in the regime where error-correction protocols become inefficient.

quant-ph↗

arXiv2024

Intrinsic Dimension Correlation: uncovering nonlinear connections in multimodal representations

To gain insight into the mechanisms behind machine learning methods, it is crucial to establish connections among the features describing data points. However, these correlations often exhibit a high-dimensional and strongly nonlinear nature, which makes them challenging to detect using standard methods. This paper exploits the entanglement between intrinsic dimensionality and correlation to propose a metric that quantifies the (potentially nonlinear) correlation between high-dimensional manifolds. We first validate our method on synthetic data in controlled environments, showcasing its advantages and drawbacks compared to existing techniques. Subsequently, we extend our analysis to large-scale applications in neural network representations. Specifically, we focus on latent representations of multimodal data, uncovering clear correlations between paired visual and textual embeddings, whereas existing methods struggle significantly in detecting similarity. Our results indicate the presence of highly nonlinear correlation patterns between latent manifolds.

cs.LG↗

arXiv2023

Machine Learning Catalysis of Quantum Tunneling

Optimizing the probability of quantum tunneling between two states, while keeping the resources of the underlying physical system constant, is a task of key importance due to its critical role in various applications. We show that, by applying Machine Learning techniques when the system is coupled to an ancilla, one optimizes the parameters of both the ancillary component and the coupling, ultimately resulting in the maximization of the tunneling probability. We provide illustrative examples for the paradigmatic scenario involving a two-mode system and a two-mode ancilla in the presence of several interacting particles. Physically, the increase of the tunneling probability is rooted in the decrease of the two-well asymmetry due to the coherent oscillations induced by the coupling to the ancilla. We also argue that the enhancement of the tunneling probability is not hampered by weak coupling to noisy environments.

quant-ph↗

arXiv2023

Catalysis of quantum tunneling by ancillary system learning

Given the key role that quantum tunneling plays in a wide range of applications, a crucial objective is to maximize the probability of tunneling from one quantum state/level to another, while keeping the resources of the underlying physical system fixed. In this work, we demonstrate that an effective solution to this challenge can be achieved by coupling the tunneling system with an ancillary system of the same kind. By utilizing machine learning techniques, the parameters of both the ancillary system and the coupling can be optimized, leading to the maximization of the tunneling probability. We provide illustrative examples for the paradigmatic scenario involving a two-mode system and a two-mode ancilla with arbitrary couplings and in the presence of several interacting particles. Importantly, the enhancement of the tunneling probability appears to be minimally affected by noise and decoherence in both the system and the ancilla.

quant-ph↗

arXiv2023

Local Search, Semantics, and Genetic Programming: a Global Analysis

Geometric Semantic Geometric Programming (GSGP) is one of the most prominent Genetic Programming (GP) variants, thanks to its solid theoretical background, the excellent performance achieved, and the execution time significantly smaller than standard syntax-based GP. In recent years, a new mutation operator, Geometric Semantic Mutation with Local Search (GSM-LS), has been proposed to include a local search step in the mutation process based on the idea that performing a linear regression during the mutation can allow for a faster convergence to good-quality solutions. While GSM-LS helps the convergence of the evolutionary search, it is prone to overfitting. Thus, it was suggested to use GSM-LS only for a limited number of generations and, subsequently, to switch back to standard geometric semantic mutation. A more recently defined variant of GSGP (called GSGP-reg) also includes a local search step but shares similar strengths and weaknesses with GSM-LS. Here we explore multiple possibilities to limit the overfitting of GSM-LS and GSGP-reg, ranging from adaptive methods to estimate the risk of overfitting at each mutation to a simple regularized regression. The results show that the method used to limit overfitting is not that important: providing that a technique to control overfitting is used, it is possible to consistently outperform standard GSGP on both training and unseen data. The obtained results allow practitioners to better understand the role of local search in GSGP and demonstrate that simple regularization strategies are effective in controlling overfitting.

cs.NE↗

arXiv2023

Frequency maps reveal the correlation between Adversarial Attacks and Implicit Bias

Despite their impressive performance in classification tasks, neural networks are known to be vulnerable to adversarial attacks, subtle perturbations of the input data designed to deceive the model. In this work, we investigate the correlation between these perturbations and the implicit bias of neural networks trained with gradient-based algorithms. To this end, we analyse a representation of the network's implicit bias through the lens of the Fourier transform. Specifically, we identify unique fingerprints of implicit bias and adversarial attacks by calculating the minimal, essential frequencies needed for accurate classification of each image, as well as the frequencies that drive misclassification in its adversarially perturbed counterpart. This approach enables us to uncover and analyse the correlation between these essential frequencies, providing a precise map of how the network's biases align or contrast with the frequency components exploited by adversarial attacks. To this end, among other methods, we use a newly introduced technique capable of detecting nonlinear correlations between high-dimensional datasets. Our results provide empirical evidence that the network bias in Fourier space and the target frequencies of adversarial attacks are highly correlated and suggest new potential strategies for adversarial defence.

cs.LG↗

arXiv2023

Local Convolutions Cause an Implicit Bias towards High Frequency Adversarial Examples

Adversarial Attacks are still a significant challenge for neural networks. Recent work has shown that adversarial perturbations typically contain high-frequency features, but the root cause of this phenomenon remains unknown. Inspired by theoretical work on linear full-width convolutional models, we hypothesize that the local (i.e. bounded-width) convolutional operations commonly used in current neural networks are implicitly biased to learn high frequency features, and that this is one of the root causes of high frequency adversarial examples. To test this hypothesis, we analyzed the impact of different choices of linear and nonlinear architectures on the implicit bias of the learned features and the adversarial perturbations, in both spatial and frequency domains. We find that the high-frequency adversarial perturbations are critically dependent on the convolution operation because the spatially-limited nature of local convolutions induces an implicit bias towards high frequency features. The explanation for the latter involves the Fourier Uncertainty Principle: a spatially-limited (local in the space domain) filter cannot also be frequency-limited (local in the frequency domain). Furthermore, using larger convolution kernel sizes or avoiding convolutions (e.g. by using Vision Transformers architecture) significantly reduces this high frequency bias, but not the overall susceptibility to attacks. Looking forward, our work strongly suggests that understanding and controlling the implicit bias of architectures will be essential for achieving adversarial robustness.

stat.ML↗

arXiv2022

Interpreting convolutional neural networks' low dimensional approximation to quantum spin systems

Convolutional neural networks (CNNs) have been employed along with Variational Monte Carlo methods for finding the ground state of quantum many-body spin systems with great success. In order to do so, however, a CNN with only linearly many variational parameters has to circumvent the ``curse of dimensionality'' and successfully approximate a wavefunction on an exponentially large Hilbert space. In our work, we provide a theoretical and experimental analysis of how the CNN optimizes learning for spin systems, and investigate the CNN's low dimensional approximation. We first quantify the role played by physical symmetries of the underlying spin system during training. We incorporate our insights into a new training algorithm and demonstrate its improved efficiency, accuracy and robustness. We then further investigate the CNN's ability to approximate wavefunctions by looking at the entanglement spectrum captured by the size of the convolutional filter. Our insights reveal the CNN to be an ansatz fundamentally centered around the occurrence statistics of $K$-motifs of the input strings. We use this motivation to provide the shallow CNN ansatz with a unifying theoretical interpretation in terms of other well-known statistical and physical ansatzes such as the maximum entropy (MaxEnt) and entangled plaquette correlator product states (EP-CPS). Using regression analysis, we find further relationships between the CNN's approximations of the different motifs' expectation values. Our results allow us to gain a comprehensive, improved understanding of how CNNs successfully approximate quantum spin Hamiltonians and to use that understanding to improve CNN performance.

quant-ph↗

arXiv2022

Understanding robustness and generalization of artificial neural networks through Fourier masks

Despite the enormous success of artificial neural networks (ANNs) in many disciplines, the characterization of their computations and the origin of key properties such as generalization and robustness remain open questions. Recent literature suggests that robust networks with good generalization properties tend to be biased towards processing low frequencies in images. To explore the frequency bias hypothesis further, we develop an algorithm that allows us to learn modulatory masks highlighting the essential input frequencies needed for preserving a trained network's performance. We achieve this by imposing invariance in the loss with respect to such modulations in the input frequencies. We first use our method to test the low-frequency preference hypothesis of adversarially trained or data-augmented networks. Our results suggest that adversarially robust networks indeed exhibit a low-frequency bias but we find this bias is also dependent on directions in frequency space. However, this is not necessarily true for other types of data augmentation. Our results also indicate that the essential frequencies in question are effectively the ones used to achieve generalization in the first place. Surprisingly, images seen through these modulatory masks are not recognizable and resemble texture-like patterns.

cs.CV↗

arXiv2019

A computational model for grid maps in neural populations

Grid cells in the entorhinal cortex, together with head direction, place, speed and border cells, are major contributors to the organization of spatial representations in the brain. In this work we introduce a novel theoretical and algorithmic framework able to explain the emergence of hexagonal grid-like response patterns from head direction cells' responses. We show that this pattern is a result of minimal variance encoding of neurons. The novelty lies into the formulation of the encoding problem through the modern Frame Theory language, specifically that of equiangular Frames, providing new insights about the optimality of hexagonal grid receptive fields. The model proposed overcomes some crucial limitations of the current attractor and oscillatory models. It is based on the well-accepted and tested hypothesis of Hebbian learning, providing a simplified cortical-based framework that does not require the presence of theta velocity-driven oscillations (oscillatory model) or translational symmetries in the synaptic connections (attractor model). We moreover demonstrate that the proposed encoding mechanism naturally explains axis alignment of neighbor grid cells and maps shifts, rotations and scaling of the stimuli onto the shape of grid cells' receptive fields, giving a straightforward explanation of the experimental evidence of grid cells remapping under transformations of environmental cues.

q-bio.NC↗