Texture Representations in Deep Vision Models: Comparing CNNs, Vision Transformers, and Human Perception

Authors: Ludovica de Paolis, Marco Baroni, Alessandro Laio, Eugenio Piasini

Abstract: In computational vision science, Convolutional Neural Networks (CNNs) have emerged as a popular model of biological vision because of the alignment they can exhibit with neural and behavioral data in humans and animals. However, it remains unclear to what extent this alignment persists for visual tasks that extend beyond the canonical object recognition paradigm based on well defined semantic cont… ▽ More In computational vision science, Convolutional Neural Networks (CNNs) have emerged as a popular model of biological vision because of the alignment they can exhibit with neural and behavioral data in humans and animals. However, it remains unclear to what extent this alignment persists for visual tasks that extend beyond the canonical object recognition paradigm based on well defined semantic content. In this study, we diverge from the common object-centric view by focusing on another aspect of vision: texture perception. We consider textures of different complexity generated with three different algorithms from the same source images. Using a rank-based statistic, we quantify the information encoded in the internal representations of a CNN and three Vision Transformers (ViTs), and we compare the similarity of these representations to those inferred from human psychophysics data. We find that the representation of textures is aligned in different ViTs, but not between the ViTs and the CNN; that ViTs form similar representations for textures of different complexity; that human performance in recognizing textures can be better predicted from ViTs representations rather than CNN representations. Taken together, these results suggest that ViTs may capture more faithfully than CNNs how texture patterns are visually processed by humans, and that the representations of texture stimuli in computational models may be driven by the network architecture. △ Less

Submitted 9 July, 2026; originally announced July 2026.

Signal-to-Noise Ratio and Sample Size Govern Representational Alignment in Neural Networks

Authors: Ali Hussaini Umar, Alessandro Laio

Abstract: Neural networks are known to develop latent representations that are $aligned$, namely structurally similar across networks trained with different architectures, training protocols, or training datasets. We study this phenomenon in a controlled setting, where we train an ensemble of networks on regression and classification tasks using training sets perturbed by independent realizations of a noise… ▽ More Neural networks are known to develop latent representations that are $aligned$, namely structurally similar across networks trained with different architectures, training protocols, or training datasets. We study this phenomenon in a controlled setting, where we train an ensemble of networks on regression and classification tasks using training sets perturbed by independent realizations of a noise process. We show that the signal-to-noise ratio (SNR) and the training sample size influence the alignment in qualitatively similar ways in networks trained on real-world datasets and in an extremely simple $linear$ network with a single hidden layer, for which the alignment can be estimated analytically. Across linear and nonlinear networks, regression and classification tasks, and both synthetic and real-world data, we consistently observe that alignment varies monotonically with SNR but non-monotonically with training sample size. In particular, the alignment is minimized near the interpolation threshold, and a stronger alignment does not necessarily correspond to better generalization error. These findings reveal a non-trivial dependence of alignment on data quality and quantity, decoupled from generalization performance. △ Less

Submitted 26 May, 2026; originally announced May 2026.

Unsupervised Domain Shift Detection with Interpretable Subspace Attribution

Authors: Sebastian Springer, Alessandro Laio

Abstract: We developed a tool for detecting domain shifts, namely subtle differences in the probability distributions of datasets. We identify these shifts using an algorithm designed to detect localised density anomalies in high-dimensional feature spaces. If an anomaly is present, we then identify the feature subspace in which the anomaly is most pronounced. This allows us to trace the domain shift to a s… ▽ More We developed a tool for detecting domain shifts, namely subtle differences in the probability distributions of datasets. We identify these shifts using an algorithm designed to detect localised density anomalies in high-dimensional feature spaces. If an anomaly is present, we then identify the feature subspace in which the anomaly is most pronounced. This allows us to trace the domain shift to a small set of features, making the shift interpretable. Moreover, we provide a protocol for compensating domain shifts by extracting, from two unlabelled datasets, subsets of samples with no detectable residual distributional difference. We validate the framework on controlled 20-dimensional benchmarks with known ground truth, recovering both broad and localized shifts together with their supporting feature subspaces. We then apply it to healthy electrocardiogram (ECG) recordings represented by 782 features. In age- and sex-matched cohort comparisons differing in measurement-device composition, the method detects device-induced shifts, extracts representative subsets enriched in the imbalanced device components, and identifies ECG features associated with the acquisition contrast. These results suggest that density-shift detection and subspace attribution provide a practical framework for uncovering hidden cohort biases before downstream modelling. △ Less

Submitted 15 May, 2026; originally announced May 2026.

Differential syntactic and semantic encoding in LLMs

Authors: Santiago Acevedo, Alessandro Laio, Marco Baroni

Abstract: We study how syntactic and semantic information is encoded in inner layer representations of Large Language Models (LLMs), focusing on the very large DeepSeek-V3. We find that, by averaging hidden-representation vectors of sentences sharing syntactic structure or meaning, we obtain vectors that capture a significant proportion of the syntactic and semantic information contained in the representati… ▽ More We study how syntactic and semantic information is encoded in inner layer representations of Large Language Models (LLMs), focusing on the very large DeepSeek-V3. We find that, by averaging hidden-representation vectors of sentences sharing syntactic structure or meaning, we obtain vectors that capture a significant proportion of the syntactic and semantic information contained in the representations. In particular, subtracting these syntactic and semantic ``centroids'' from sentence vectors strongly affects their similarity with syntactically and semantically matched sentences, respectively, suggesting that syntax and semantics are, at least partially, linearly encoded. We also find that the cross-layer encoding profiles of syntax and semantics are different, and that the two signals can to some extent be decoupled, suggesting differential encoding of these two types of linguistic information in LLM representations. △ Less

Submitted 28 May, 2026; v1 submitted 8 January, 2026; originally announced January 2026.

The Effect of Label Noise on the Information Content of Neural Representations

Authors: Ali Hussaini Umar, Franky Kevin Nando Tezoh, Jean Barbier, Santiago Acevedo, Alessandro Laio

Abstract: In supervised classification tasks, models are trained to predict a label for each data point. In real-world datasets, these labels are often noisy due to annotation errors. While the impact of label noise on the performance of deep learning models has been widely studied, its effects on the networks' hidden representations remain poorly understood. We address this gap by systematically comparing… ▽ More In supervised classification tasks, models are trained to predict a label for each data point. In real-world datasets, these labels are often noisy due to annotation errors. While the impact of label noise on the performance of deep learning models has been widely studied, its effects on the networks' hidden representations remain poorly understood. We address this gap by systematically comparing hidden representations using the Information Imbalance, a computationally efficient proxy of conditional mutual information. Through this analysis, we observe that the information content of the hidden representations follows a double descent as a function of the number of network parameters, akin to the behavior of the test error. We further demonstrate that in the underparameterized regime, representations learned with noisy labels are more informative than those learned with clean labels, while in the overparameterized regime, these representations are equally informative. Our results indicate that the representations of overparameterized networks are robust to label noise. We also found that the information imbalance between the penultimate and pre-softmax layers decreases with cross-entropy loss in the overparameterized regime. This offers a new perspective on understanding generalization in classification tasks. Extending our analysis to representations learned from random labels, we show that these perform worse than random features. This indicates that training on random labels drives networks much beyond lazy learning, as weights adapt to encode labels information. △ Less

Submitted 7 October, 2025; originally announced October 2025.

A quantitative analysis of semantic information in deep representations of text and images

Authors: Santiago Acevedo, Andrea Mascaretti, Riccardo Rende, Matéo Mahaut, Marco Baroni, Alessandro Laio

Abstract: It was recently observed that the representations of different models that process identical or semantically related inputs tend to align. We analyze this phenomenon using the Information Imbalance, an asymmetric rank-based measure that quantifies the capability of a representation to predict another, providing a proxy of the cross-entropy which can be computed efficiently in high-dimensional spac… ▽ More It was recently observed that the representations of different models that process identical or semantically related inputs tend to align. We analyze this phenomenon using the Information Imbalance, an asymmetric rank-based measure that quantifies the capability of a representation to predict another, providing a proxy of the cross-entropy which can be computed efficiently in high-dimensional spaces. By measuring the Information Imbalance between representations generated by DeepSeek-V3 processing translations, we find that semantic information is spread across many tokens, and that semantic predictability is strongest in a set of central layers of the network, robust across six language pairs. We measure clear information asymmetries: English representations are systematically more predictive than those of other languages, and DeepSeek-V3 representations are more predictive of those in a smaller model such as Llama3-8b than the opposite. In the visual domain, we observe that semantic information concentrates in middle layers for autoregressive models and in final layers for encoder models, and these same layers yield the strongest cross-modal predictability with textual representations of image captions. Our results support the hypothesis of semantic convergence across languages, modalities, and architectures, while showing that directed predictability between representations varies strongly with layer-depth, model scale, and language. △ Less

Submitted 6 July, 2026; v1 submitted 21 May, 2025; originally announced May 2025.

Detecting Localized Density Anomalies in Multivariate Data via Coin-Flip Statistics

Authors: Sebastian Springer, Andre Scaffidi, Maximilian Autenrieth, Gabriella Contardo, Alessandro Laio, Roberto Trotta, Heikki Haario

Abstract: Detecting localized differences between two samples is a central task in scientific data analysis, required for the identification of signal events, regime changes, or model mismatch. We introduce EagleEye, a method that pinpoints local over- and under-densities in multivariate feature spaces. EagleEye assigns each point an anomaly score by encoding its ordered k-nearest-neighbour list as a binary… ▽ More Detecting localized differences between two samples is a central task in scientific data analysis, required for the identification of signal events, regime changes, or model mismatch. We introduce EagleEye, a method that pinpoints local over- and under-densities in multivariate feature spaces. EagleEye assigns each point an anomaly score by encoding its ordered k-nearest-neighbour list as a binary membership sequence and testing whether the cumulative number of successes in this sequence is consistent with a binomial (coin-flipping) null model. In the presence of a genuine local anomaly, neighbours will preferentially belong to one of the two datasts, yielding an excess of ``successes'' relative to the binomial null model. These local, pointwise detections are consolidated into interpretable anomaly sets through a deterministic refinement procedure that can also estimate the irreducible background and local density anomaly purity. We demonstrate EagleEye's efficacy in three scenarios. We first consider an artificial data example with known localized over- and under-densities. Second, we demonstrate how EagleEye may be used for new physics searches at particle collider experiments in the presence of systematic background modelling differences. Finally, we conduct a climate analysis study that reveals localized changes in spatiotemporal temperature-pattern recurrence. △ Less

Submitted 15 May, 2026; v1 submitted 31 March, 2025; originally announced March 2025.

Unsupervised detection of semantic correlations in big data

Authors: Santiago Acevedo, Alex Rodriguez, Alessandro Laio

Abstract: In real-world data, information is stored in extremely large feature vectors. These variables are typically correlated due to complex interactions involving many features simultaneously. Such correlations qualitatively correspond to semantic roles and are naturally recognized by both the human brain and artificial neural networks. This recognition enables, for instance, the prediction of missing p… ▽ More In real-world data, information is stored in extremely large feature vectors. These variables are typically correlated due to complex interactions involving many features simultaneously. Such correlations qualitatively correspond to semantic roles and are naturally recognized by both the human brain and artificial neural networks. This recognition enables, for instance, the prediction of missing parts of an image or text based on their context. We present a method to detect these correlations in high-dimensional data represented as binary numbers. We estimate the binary intrinsic dimension of a dataset, which quantifies the minimum number of independent coordinates needed to describe the data, and is therefore a proxy of semantic complexity. The proposed algorithm is largely insensitive to the so-called curse of dimensionality, and can therefore be used in big data analysis. We test this approach identifying phase transitions in model magnetic systems and we then apply it to the detection of semantic correlations of images and text inside deep neural networks. △ Less

Submitted 21 May, 2025; v1 submitted 4 November, 2024; originally announced November 2024.

Automatic feature selection and weighting in molecular systems using Differentiable Information Imbalance

Authors: Romina Wild, Felix Wodaczek, Vittorio Del Tatto, Bingqing Cheng, Alessandro Laio

Abstract: Feature selection is essential in the analysis of molecular systems and many other fields, but several uncertainties remain: What is the optimal number of features for a simplified, interpretable model that retains essential information? How should features with different units be aligned, and how should their relative importance be weighted? Here, we introduce the Differentiable Information Imbal… ▽ More Feature selection is essential in the analysis of molecular systems and many other fields, but several uncertainties remain: What is the optimal number of features for a simplified, interpretable model that retains essential information? How should features with different units be aligned, and how should their relative importance be weighted? Here, we introduce the Differentiable Information Imbalance (DII), an automated method to rank information content between sets of features. Using distances in a ground truth feature space, DII identifies a low-dimensional subset of features that best preserves these relationships. Each feature is scaled by a weight, which is optimized by minimizing the DII through gradient descent. This allows simultaneously performing unit alignment and relative importance scaling, while preserving interpretability. DII can also produce sparse solutions and determine the optimal size of the reduced feature space. We demonstrate the usefulness of this approach on two benchmark molecular problems: (1) identifying collective variables that describe conformations of a biomolecule, and (2) selecting features for training a machine-learning force field. These results show the potential of DII in addressing feature selection challenges and optimizing dimensionality in various applications. The method is available in the Python library DADApy. △ Less

Submitted 30 December, 2024; v1 submitted 30 October, 2024; originally announced November 2024.

A distributional simplicity bias in the learning dynamics of transformers

Authors: Riccardo Rende, Federica Gerace, Alessandro Laio, Sebastian Goldt

Abstract: The remarkable capability of over-parameterised neural networks to generalise effectively has been explained by invoking a ``simplicity bias'': neural networks prevent overfitting by initially learning simple classifiers before progressing to more complex, non-linear functions. While simplicity biases have been described theoretically and experimentally in feed-forward networks for supervised lear… ▽ More The remarkable capability of over-parameterised neural networks to generalise effectively has been explained by invoking a ``simplicity bias'': neural networks prevent overfitting by initially learning simple classifiers before progressing to more complex, non-linear functions. While simplicity biases have been described theoretically and experimentally in feed-forward networks for supervised learning, the extent to which they also explain the remarkable success of transformers trained with self-supervised techniques remains unclear. In our study, we demonstrate that transformers, trained on natural language data, also display a simplicity bias. Specifically, they sequentially learn many-body interactions among input tokens, reaching a saturation point in the prediction error for low-degree interactions while continuing to learn high-degree interactions. To conduct this analysis, we develop a procedure to generate \textit{clones} of a given natural language data set, which rigorously capture the interactions between tokens up to a specified order. This approach opens up the possibilities of studying how interactions of different orders in the data affect learning, in natural language processing and beyond. △ Less

Submitted 14 February, 2025; v1 submitted 25 October, 2024; originally announced October 2024.

Density Estimation via Binless Multidimensional Integration

Authors: Matteo Carli, Alex Rodriguez, Alessandro Laio, Aldo Glielmo

Abstract: We introduce the Binless Multidimensional Thermodynamic Integration (BMTI) method for nonparametric, robust, and data-efficient density estimation. BMTI estimates the logarithm of the density by initially computing log-density differences between neighbouring data points. Subsequently, such differences are integrated, weighted by their associated uncertainties, using a maximum-likelihood formulati… ▽ More We introduce the Binless Multidimensional Thermodynamic Integration (BMTI) method for nonparametric, robust, and data-efficient density estimation. BMTI estimates the logarithm of the density by initially computing log-density differences between neighbouring data points. Subsequently, such differences are integrated, weighted by their associated uncertainties, using a maximum-likelihood formulation. This procedure can be seen as an extension to a multidimensional setting of the thermodynamic integration, a technique developed in statistical physics. The method leverages the manifold hypothesis, estimating quantities within the intrinsic data manifold without defining an explicit coordinate map. It does not rely on any binning or space partitioning, but rather on the construction of a neighbourhood graph based on an adaptive bandwidth selection procedure. BMTI mitigates the limitations commonly associated with traditional nonparametric density estimators, effectively reconstructing smooth profiles even in high-dimensional embedding spaces. The method is tested on a variety of complex synthetic high-dimensional datasets, where it is shown to outperform traditional estimators, and is benchmarked on realistic datasets from the chemical physics literature. △ Less

Submitted 14 May, 2026; v1 submitted 10 July, 2024; originally announced July 2024.

Emergence of a High-Dimensional Abstraction Phase in Language Transformers

Authors: Emily Cheng, Diego Doimo, Corentin Kervadec, Iuri Macocco, Jade Yu, Alessandro Laio, Marco Baroni

Abstract: A language model (LM) is a mapping from a linguistic context to an output token. However, much remains to be known about this mapping, including how its geometric properties relate to its function. We take a high-level geometric approach to its analysis, observing, across five pre-trained transformer-based LMs and three input datasets, a distinct phase characterized by high intrinsic dimensionalit… ▽ More A language model (LM) is a mapping from a linguistic context to an output token. However, much remains to be known about this mapping, including how its geometric properties relate to its function. We take a high-level geometric approach to its analysis, observing, across five pre-trained transformer-based LMs and three input datasets, a distinct phase characterized by high intrinsic dimensionality. During this phase, representations (1) correspond to the first full linguistic abstraction of the input; (2) are the first to viably transfer to downstream tasks; (3) predict each other across different LMs. Moreover, we find that an earlier onset of the phase strongly predicts better language modelling performance. In short, our results suggest that a central high-dimensionality phase underlies core linguistic processing in many common LM architectures. △ Less

Submitted 30 April, 2025; v1 submitted 24 May, 2024; originally announced May 2024.

Scale-adaptive and robust intrinsic dimension estimation via optimal neighbourhood identification

Authors: Antonio Di Noia, Iuri Macocco, Aldo Glielmo, Alessandro Laio, Antonietta Mira

Abstract: The Intrinsic Dimension (ID) is a key concept in unsupervised learning and feature selection, as it is a lower bound to the number of variables which are necessary to describe a system. However, in almost any real-world dataset the ID depends on the scale at which the data are analysed. Quite typically at a small scale, the ID is very large, as the data are affected by measurement errors. At large… ▽ More The Intrinsic Dimension (ID) is a key concept in unsupervised learning and feature selection, as it is a lower bound to the number of variables which are necessary to describe a system. However, in almost any real-world dataset the ID depends on the scale at which the data are analysed. Quite typically at a small scale, the ID is very large, as the data are affected by measurement errors. At large scale, the ID can also appear erroneously large, due to the curvature and the topology of the manifold containing the data. In this work, we introduce an automatic protocol to select the sweet spot, namely the correct range of scales in which the ID is meaningful and useful. This protocol is based on imposing that for distances smaller than the correct scale the density of the data is constant. In the presented framework, to estimate the density it is necessary to know the ID, therefore, this condition is imposed self-consistently. We illustrate the usefulness and robustness of this procedure to noise by benchmarks on artificial and real-world datasets. △ Less

Submitted 1 April, 2026; v1 submitted 23 May, 2024; originally announced May 2024.

Mapping of attention mechanisms to a generalized Potts model

Authors: Riccardo Rende, Federica Gerace, Alessandro Laio, Sebastian Goldt

Abstract: Transformers are neural networks that revolutionized natural language processing and machine learning. They process sequences of inputs, like words, using a mechanism called self-attention, which is trained via masked language modeling (MLM). In MLM, a word is randomly masked in an input sequence, and the network is trained to predict the missing word. Despite the practical success of transformers… ▽ More Transformers are neural networks that revolutionized natural language processing and machine learning. They process sequences of inputs, like words, using a mechanism called self-attention, which is trained via masked language modeling (MLM). In MLM, a word is randomly masked in an input sequence, and the network is trained to predict the missing word. Despite the practical success of transformers, it remains unclear what type of data distribution self-attention can learn efficiently. Here, we show analytically that if one decouples the treatment of word positions and embeddings, a single layer of self-attention learns the conditionals of a generalized Potts model with interactions between sites and Potts colors. Moreover, we show that training this neural network is exactly equivalent to solving the inverse Potts problem by the so-called pseudo-likelihood method, well known in statistical physics. Using this mapping, we compute the generalization error of self-attention in a model scenario analytically using the replica method. △ Less

Submitted 4 April, 2024; v1 submitted 14 April, 2023; originally announced April 2023.

Optimal transfer protocol by incremental layer defrosting

Authors: Federica Gerace, Diego Doimo, Stefano Sarao Mannelli, Luca Saglietti, Alessandro Laio

Abstract: Transfer learning is a powerful tool enabling model training with limited amounts of data. This technique is particularly useful in real-world problems where data availability is often a serious limitation. The simplest transfer learning protocol is based on ``freezing" the feature-extractor layers of a network pre-trained on a data-rich source task, and then adapting only the last layers to a dat… ▽ More Transfer learning is a powerful tool enabling model training with limited amounts of data. This technique is particularly useful in real-world problems where data availability is often a serious limitation. The simplest transfer learning protocol is based on ``freezing" the feature-extractor layers of a network pre-trained on a data-rich source task, and then adapting only the last layers to a data-poor target task. This workflow is based on the assumption that the feature maps of the pre-trained model are qualitatively similar to the ones that would have been learned with enough data on the target task. In this work, we show that this protocol is often sub-optimal, and the largest performance gain may be achieved when smaller portions of the pre-trained network are kept frozen. In particular, we make use of a controlled framework to identify the optimal transfer depth, which turns out to depend non-trivially on the amount of available training data and on the degree of source-target task correlation. We then characterize transfer optimality by analyzing the internal representations of two networks trained from scratch on the source and the target task through multiple established similarity measures. △ Less

Submitted 2 March, 2023; originally announced March 2023.

The geometry of hidden representations of large transformer models

Authors: Lucrezia Valeriani, Diego Doimo, Francesca Cuturello, Alessandro Laio, Alessio Ansuini, Alberto Cazzaniga

Abstract: Large transformers are powerful architectures used for self-supervised data analysis across various data types, including protein sequences, images, and text. In these models, the semantic structure of the dataset emerges from a sequence of transformations between one representation and the next. We characterize the geometric and statistical properties of these representations and how they change… ▽ More Large transformers are powerful architectures used for self-supervised data analysis across various data types, including protein sequences, images, and text. In these models, the semantic structure of the dataset emerges from a sequence of transformations between one representation and the next. We characterize the geometric and statistical properties of these representations and how they change as we move through the layers. By analyzing the intrinsic dimension (ID) and neighbor composition, we find that the representations evolve similarly in transformers trained on protein language tasks and image reconstruction tasks. In the first layers, the data manifold expands, becoming high-dimensional, and then contracts significantly in the intermediate layers. In the last part of the model, the ID remains approximately constant or forms a second shallow peak. We show that the semantic information of the dataset is better expressed at the end of the first peak, and this phenomenon can be observed across many models trained on diverse datasets. Based on our findings, we point out an explicit strategy to identify, without supervision, the layers that maximize semantic content: representations at intermediate layers corresponding to a relative minimum of the ID profile are more suitable for downstream learning tasks. △ Less

Submitted 30 October, 2023; v1 submitted 1 February, 2023; originally announced February 2023.

Intrinsic dimension estimation for discrete metrics

Authors: Iuri Macocco, Aldo Glielmo, Jacopo Grilli, Alessandro Laio

Abstract: Real world-datasets characterized by discrete features are ubiquitous: from categorical surveys to clinical questionnaires, from unweighted networks to DNA sequences. Nevertheless, the most common unsupervised dimensional reduction methods are designed for continuous spaces, and their use for discrete spaces can lead to errors and biases. In this letter we introduce an algorithm to infer the intri… ▽ More Real world-datasets characterized by discrete features are ubiquitous: from categorical surveys to clinical questionnaires, from unweighted networks to DNA sequences. Nevertheless, the most common unsupervised dimensional reduction methods are designed for continuous spaces, and their use for discrete spaces can lead to errors and biases. In this letter we introduce an algorithm to infer the intrinsic dimension (ID) of datasets embedded in discrete spaces. We demonstrate its accuracy on benchmark datasets, and we apply it to analyze a metagenomic dataset for species fingerprinting, finding a surprisingly small ID, of order 2. This suggests that evolutive pressure acts on a low-dimensional manifold despite the high-dimensionality of sequences' space. △ Less

Submitted 12 March, 2023; v1 submitted 20 July, 2022; originally announced July 2022.

Prune and distill: similar reformatting of image information along rat visual cortex and deep neural networks

Authors: Paolo Muratore, Sina Tafazoli, Eugenio Piasini, Alessandro Laio, Davide Zoccolan

Abstract: Visual object recognition has been extensively studied in both neuroscience and computer vision. Recently, the most popular class of artificial systems for this task, deep convolutional neural networks (CNNs), has been shown to provide excellent models for its functional analogue in the brain, the ventral stream in visual cortex. This has prompted questions on what, if any, are the common principl… ▽ More Visual object recognition has been extensively studied in both neuroscience and computer vision. Recently, the most popular class of artificial systems for this task, deep convolutional neural networks (CNNs), has been shown to provide excellent models for its functional analogue in the brain, the ventral stream in visual cortex. This has prompted questions on what, if any, are the common principles underlying the reformatting of visual information as it flows through a CNN or the ventral stream. Here we consider some prominent statistical patterns that are known to exist in the internal representations of either CNNs or the visual cortex and look for them in the other system. We show that intrinsic dimensionality (ID) of object representations along the rat homologue of the ventral stream presents two distinct expansion-contraction phases, as previously shown for CNNs. Conversely, in CNNs, we show that training results in both distillation and active pruning (mirroring the increase in ID) of low- to middle-level image information in single units, as representations gain the ability to support invariant discrimination, in agreement with previous observations in rat visual cortex. Taken together, our findings suggest that CNNs and visual cortex share a similarly tight relationship between dimensionality expansion/reduction of object representations and reformatting of image information. △ Less

Submitted 27 May, 2022; originally announced May 2022.

DADApy: Distance-based Analysis of DAta-manifolds in Python

Authors: Aldo Glielmo, Iuri Macocco, Diego Doimo, Matteo Carli, Claudio Zeni, Romina Wild, Maria d'Errico, Alex Rodriguez, Alessandro Laio

Abstract: DADApy is a python software package for analysing and characterising high-dimensional data manifolds. It provides methods for estimating the intrinsic dimension and the probability density, for performing density-based clustering and for comparing different distance metrics. We review the main functionalities of the package and exemplify its usage in toy cases and in a real-world application. DADA… ▽ More DADApy is a python software package for analysing and characterising high-dimensional data manifolds. It provides methods for estimating the intrinsic dimension and the probability density, for performing density-based clustering and for comparing different distance metrics. We review the main functionalities of the package and exemplify its usage in toy cases and in a real-world application. DADApy is freely available under the open-source Apache 2.0 license. △ Less

Submitted 19 September, 2022; v1 submitted 4 May, 2022; originally announced May 2022.

Redundant representations help generalization in wide neural networks

Authors: Diego Doimo, Aldo Glielmo, Sebastian Goldt, Alessandro Laio

Abstract: Deep neural networks (DNNs) defy the classical bias-variance trade-off: adding parameters to a DNN that interpolates its training data will typically improve its generalization performance. Explaining the mechanism behind this ``benign overfitting'' in deep networks remains an outstanding challenge. Here, we study the last hidden layer representations of various state-of-the-art convolutional neur… ▽ More Deep neural networks (DNNs) defy the classical bias-variance trade-off: adding parameters to a DNN that interpolates its training data will typically improve its generalization performance. Explaining the mechanism behind this ``benign overfitting'' in deep networks remains an outstanding challenge. Here, we study the last hidden layer representations of various state-of-the-art convolutional neural networks and find that if the last hidden representation is wide enough, its neurons tend to split into groups that carry identical information, and differ from each other only by statistically independent noise. The number of such groups increases linearly with the width of the layer, but only if the width is above a critical value. We show that redundant neurons appear only when the training process reaches interpolation and the training error is zero. △ Less

Submitted 29 April, 2023; v1 submitted 7 June, 2021; originally announced June 2021.

Ranking the information content of distance measures

Authors: Aldo Glielmo, Claudio Zeni, Bingqing Cheng, Gabor Csanyi, Alessandro Laio

Abstract: Real-world data typically contain a large number of features that are often heterogeneous in nature, relevance, and also units of measure. When assessing the similarity between data points, one can build various distance measures using subsets of these features. Using the fewest features but still retaining sufficient information about the system is crucial in many statistical learning approaches,… ▽ More Real-world data typically contain a large number of features that are often heterogeneous in nature, relevance, and also units of measure. When assessing the similarity between data points, one can build various distance measures using subsets of these features. Using the fewest features but still retaining sufficient information about the system is crucial in many statistical learning approaches, particularly when data are sparse. We introduce a statistical test that can assess the relative information retained when using two different distance measures, and determine if they are equivalent, independent, or if one is more informative than the other. This in turn allows finding the most informative distance measure out of a pool of candidates. The approach is applied to find the most relevant policy variables for controlling the Covid-19 epidemic and to find compact yet informative representations of atomic structures, but its potential applications are wide ranging in many branches of science. △ Less

Submitted 25 May, 2022; v1 submitted 30 April, 2021; originally announced April 2021.

Hierarchical nucleation in deep neural networks

Authors: Diego Doimo, Aldo Glielmo, Alessio Ansuini, Alessandro Laio

Abstract: Deep convolutional networks (DCNs) learn meaningful representations where data that share the same abstract characteristics are positioned closer and closer. Understanding these representations and how they are generated is of unquestioned practical and theoretical interest. In this work we study the evolution of the probability density of the ImageNet dataset across the hidden layers in some stat… ▽ More Deep convolutional networks (DCNs) learn meaningful representations where data that share the same abstract characteristics are positioned closer and closer. Understanding these representations and how they are generated is of unquestioned practical and theoretical interest. In this work we study the evolution of the probability density of the ImageNet dataset across the hidden layers in some state-of-the-art DCNs. We find that the initial layers generate a unimodal probability density getting rid of any structure irrelevant for classification. In subsequent layers density peaks arise in a hierarchical fashion that mirrors the semantic hierarchy of the concepts. Density peaks corresponding to single categories appear only close to the output and via a very sharp transition which resembles the nucleation process of a heterogeneous liquid. This process leaves a footprint in the probability density of the output layer where the topography of the peaks allows reconstructing the semantic relationships of the categories. △ Less

Submitted 9 July, 2020; v1 submitted 7 July, 2020; originally announced July 2020.

Intrinsic dimension of data representations in deep neural networks

Authors: Alessio Ansuini, Alessandro Laio, Jakob H. Macke, Davide Zoccolan

Abstract: Deep neural networks progressively transform their inputs across multiple processing layers. What are the geometrical properties of the representations learned by these networks? Here we study the intrinsic dimensionality (ID) of data-representations, i.e. the minimal number of parameters needed to describe a representation. We find that, in a trained network, the ID is orders of magnitude smaller… ▽ More Deep neural networks progressively transform their inputs across multiple processing layers. What are the geometrical properties of the representations learned by these networks? Here we study the intrinsic dimensionality (ID) of data-representations, i.e. the minimal number of parameters needed to describe a representation. We find that, in a trained network, the ID is orders of magnitude smaller than the number of units in each layer. Across layers, the ID first increases and then progressively decreases in the final layers. Remarkably, the ID of the last hidden layer predicts classification accuracy on the test set. These results can neither be found by linear dimensionality estimates (e.g., with principal component analysis), nor in representations that had been artificially linearized. They are neither found in untrained networks, nor in networks that are trained on randomized labels. This suggests that neural networks that can generalize are those that transform the data into low-dimensional, but not necessarily flat manifolds. △ Less

Submitted 28 October, 2019; v1 submitted 29 May, 2019; originally announced May 2019.

Data segmentation based on the local intrinsic dimension

Authors: Michele Allegra, Elena Facco, Francesco Denti, Alessandro Laio, Antonietta Mira

Abstract: One of the founding paradigms of machine learning is that a small number of variables is often sufficient to describe high-dimensional data. The minimum number of variables required is called the intrinsic dimension (ID) of the data. Contrary to common intuition, there are cases where the ID varies within the same data set. This fact has been highlighted in technical discussions, but seldom exploi… ▽ More One of the founding paradigms of machine learning is that a small number of variables is often sufficient to describe high-dimensional data. The minimum number of variables required is called the intrinsic dimension (ID) of the data. Contrary to common intuition, there are cases where the ID varies within the same data set. This fact has been highlighted in technical discussions, but seldom exploited to analyze large data sets and obtain insight into their structure. Here we develop a robust approach to discriminate regions with different local IDs and segment the points accordingly. Our approach is computationally efficient and can be proficiently used even on large data sets. We find that many real-world data sets contain regions with widely heterogeneous dimensions. These regions host points differing in core properties: folded vs unfolded configurations in a protein molecular dynamics trajectory, active vs non-active regions in brain imaging data, and firms with different financial risk in company balance sheets. A simple topological feature, the local ID, is thus sufficient to achieve an unsupervised segmentation of high-dimensional data, complementary to the one given by clustering algorithms. △ Less

Submitted 13 July, 2020; v1 submitted 27 February, 2019; originally announced February 2019.

Estimating the intrinsic dimension of datasets by a minimal neighborhood information

Authors: Elena Facco, Maria d'Errico, Alex Rodriguez, Alessandro Laio

Abstract: Analyzing large volumes of high-dimensional data is an issue of fundamental importance in data science, molecular simulations and beyond. Several approaches work on the assumption that the important content of a dataset belongs to a manifold whose Intrinsic Dimension (ID) is much lower than the crude large number of coordinates. Such manifold is generally twisted and curved, in addition points on… ▽ More Analyzing large volumes of high-dimensional data is an issue of fundamental importance in data science, molecular simulations and beyond. Several approaches work on the assumption that the important content of a dataset belongs to a manifold whose Intrinsic Dimension (ID) is much lower than the crude large number of coordinates. Such manifold is generally twisted and curved, in addition points on it will be non-uniformly distributed: two factors that make the identification of the ID and its exploitation really hard. Here we propose a new ID estimator using only the distance of the first and the second nearest neighbor of each point in the sample. This extreme minimality enables us to reduce the effects of curvature, of density variation, and the resulting computational cost. The ID estimator is theoretically exact in uniformly distributed datasets, and provides consistent measures in general. When used in combination with block analysis, it allows discriminating the relevant dimensions as a function of the block size. This allows estimating the ID even when the data lie on a manifold perturbed by a high-dimensional noise, a situation often encountered in real world data sets. We demonstrate the usefulness of the approach on molecular simulations and image analysis. △ Less

Submitted 19 March, 2018; originally announced March 2018.

Automatic topography of high-dimensional data sets by non-parametric Density Peak clustering

Authors: Maria d'Errico, Elena Facco, Alessandro Laio, Alex Rodriguez

Abstract: Data analysis in high-dimensional spaces aims at obtaining a synthetic description of a data set, revealing its main structure and its salient features. We here introduce an approach providing this description in the form of a topography of the data, namely a human-readable chart of the probability density from which the data are harvested. The approach is based on an unsupervised extension of Den… ▽ More Data analysis in high-dimensional spaces aims at obtaining a synthetic description of a data set, revealing its main structure and its salient features. We here introduce an approach providing this description in the form of a topography of the data, namely a human-readable chart of the probability density from which the data are harvested. The approach is based on an unsupervised extension of Density Peak clustering and a non-parametric density estimator that measures the probability density in the manifold containing the data. This allows finding automatically the number and the height of the peaks of the probability density, and the depth of the "valleys" separating them. Importantly, the density estimator provides a measure of the error, which allows distinguishing genuine density peaks from density fluctuations due to finite sampling. The approach thus provides robust and visual information about the density peaks' height, their statistical reliability, and their hierarchical organization, offering a conceptually powerful extension of the standard clustering partitions. We show that this framework is particularly useful in the analysis of complex data sets. △ Less

Submitted 5 February, 2021; v1 submitted 28 February, 2018; originally announced February 2018.