Fabio Anselmi

Nikos Karantzas, Emma Besier, Josue Ortega Caro et al.

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.

Ludovica de Paolis, Fabio Anselmi, Alessio Ansuini et al.

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 compare 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.

Nicola Aladrah, Emanuele Ballarin, Matteo Biagetti et al.

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.

Justin Sahs, Ryan Pyle, Fabio Anselmi et al.

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.

Lorenzo Basile, Santiago Acevedo, Luca Bortolussi et al.

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.

Lorenzo Basile, Nikos Karantzas, Alberto d'Onofrio et al.

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.

Josue Ortega Caro, Yilong Ju, Ryan Pyle et al.

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.

Fabio Anselmi, Micah M. Murray, Benedetta Franceschiello

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.

Joel Z. Leibo, Qianli Liao, Winrich Freiwald et al.

The primate brain contains a hierarchy of visual areas, dubbed the ventral stream, which rapidly computes object representations that are both specific for object identity and relatively robust against identity-preserving transformations like depth-rotations. Current computational models of object recognition, including recent deep learning networks, generate these properties through a hierarchy of alternating selectivity-increasing filtering and tolerance-increasing pooling operations, similar to simple-complex cells operations. While simulations of these models recapitulate the ventral stream's progression from early view-specific to late view-tolerant representations, they fail to generate the most salient property of the intermediate representation for faces found in the brain: mirror-symmetric tuning of the neural population to head orientation. Here we prove that a class of hierarchical architectures and a broad set of biologically plausible learning rules can provide approximate invariance at the top level of the network. While most of the learning rules do not yield mirror-symmetry in the mid-level representations, we characterize a specific biologically-plausible Hebb-type learning rule that is guaranteed to generate mirror-symmetric tuning to faces tuning at intermediate levels of the architecture.

Fabio Anselmi, Lorenzo Rosasco, Cheston Tan et al.

In i-theory a typical layer of a hierarchical architecture consists of HW modules pooling the dot products of the inputs to the layer with the transformations of a few templates under a group. Such layers include as special cases the convolutional layers of Deep Convolutional Networks (DCNs) as well as the non-convolutional layers (when the group contains only the identity). Rectifying nonlinearities -- which are used by present-day DCNs -- are one of the several nonlinearities admitted by i-theory for the HW module. We discuss here the equivalence between group averages of linear combinations of rectifying nonlinearities and an associated kernel. This property implies that present-day DCNs can be exactly equivalent to a hierarchy of kernel machines with pooling and non-pooling layers. Finally, we describe a conjecture for theoretically understanding hierarchies of such modules. A main consequence of the conjecture is that hierarchies of trained HW modules minimize memory requirements while computing a selective and invariant representation.

Fabio Anselmi, Lorenzo Rosasco, Tomaso Poggio

We discuss data representation which can be learned automatically from data, are invariant to transformations, and at the same time selective, in the sense that two points have the same representation only if they are one the transformation of the other. The mathematical results here sharpen some of the key claims of i-theory -- a recent theory of feedforward processing in sensory cortex.

Fabio Anselmi, Joel Z. Leibo, Lorenzo Rosasco et al.

The present phase of Machine Learning is characterized by supervised learning algorithms relying on large sets of labeled examples ($n \to \infty$). The next phase is likely to focus on algorithms capable of learning from very few labeled examples ($n \to 1$), like humans seem able to do. We propose an approach to this problem and describe the underlying theory, based on the unsupervised, automatic learning of a ``good'' representation for supervised learning, characterized by small sample complexity ($n$). We consider the case of visual object recognition though the theory applies to other domains. The starting point is the conjecture, proved in specific cases, that image representations which are invariant to translations, scaling and other transformations can considerably reduce the sample complexity of learning. We prove that an invariant and unique (discriminative) signature can be computed for each image patch, $I$, in terms of empirical distributions of the dot-products between $I$ and a set of templates stored during unsupervised learning. A module performing filtering and pooling, like the simple and complex cells described by Hubel and Wiesel, can compute such estimates. Hierarchical architectures consisting of this basic Hubel-Wiesel moduli inherit its properties of invariance, stability, and discriminability while capturing the compositional organization of the visual world in terms of wholes and parts. The theory extends existing deep learning convolutional architectures for image and speech recognition. It also suggests that the main computational goal of the ventral stream of visual cortex is to provide a hierarchical representation of new objects/images which is invariant to transformations, stable, and discriminative for recognition---and that this representation may be continuously learned in an unsupervised way during development and visual experience.