Optimal inference of a generalised Potts model by single-layer transformers with factored attention

Transformers are the type of neural networks that has revolutionised natural language processing and protein science. Their key building block is a mechanism called self-attention which is trained to predict missing words in sentences. Despite the practical success of transformers in applications it remains unclear what self-attention learns from data, and how. Here, we give a precise analytical and numerical characterisation of transformers trained on data drawn from a generalised Potts model with interactions between sites and Potts colours. While an off-the-shelf transformer requires several layers to learn this distribution, we show analytically that a single layer of self-attention with a small modification can learn the Potts model exactly in the limit of infinite sampling. We show that this modified self-attention, that we call ``factored'', has the same functional form as the conditional probability of a Potts spin given the other spins, compute its generalisation error using the replica method from statistical physics, and derive an exact mapping to pseudo-likelihood methods for solving the inverse Ising and Potts problem.

Optimal transfer protocol by incremental layer defrosting

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.

The geometry of hidden representations of large transformer models

Large transformers are powerful architectures for self-supervised analysis of data of various nature, ranging from protein sequences to text to images. In these models, the data representation in the hidden layers live in the same space, and the semantic structure of the dataset emerges by a sequence of functionally identical transformations between one representation and the next. We here characterize the geometric and statistical properties of these representations, focusing on the evolution of such proprieties across the layers. By analyzing geometric properties such as the intrinsic dimension (ID) and the neighbor composition we find that the representations evolve in a strikingly similar manner in transformers trained on protein language tasks and image reconstruction tasks. In the first layers, the data manifold expands, becoming high-dimensional, and then it 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 complexity of the dataset emerges at the end of the first peak. This phenomenon can be observed across many models trained on diverse datasets. Based on these observations, we suggest using the ID profile as an unsupervised proxy to identify the layers which are more suitable for downstream learning tasks.

Intrinsic dimension estimation for discrete metrics

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.

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

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.

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

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. The package is freely available under the open-source Apache 2.0 license and can be downloaded from the Github page https://github.com/sissa-data-science/DADApy.

Representation mitosis in wide neural networks

Deep neural networks (DNNs) defy the classical bias-variance trade-off: adding parameters to a DNN that exactly interpolates its training data will typically improve its generalisation performance. Explaining the mechanism behind the benefit of such over-parameterisation is an outstanding challenge for deep learning theory. Here, we study the last layer representation of various deep architectures such as Wide-ResNets for image classification and find evidence for an underlying mechanism that we call *representation mitosis*: if the last hidden representation is wide enough, its neurons tend to split into groups which carry identical information, and differ from each other only by a statistically independent noise. Like in a mitosis process, the number of such groups, or ``clones'', increases linearly with the width of the layer, but only if the width is above a critical value. We show that a key ingredient to activate mitosis is continuing the training process until the training error is zero. Finally, we show that in one of the learning tasks we considered, a wide model with several automatically developed clones performs significantly better than a deep ensemble based on architectures in which the last layer has the same size as the clones.

Ranking the information content of distance measures

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.

Dynamical Landscape and Multistability of the Earth's Climate

We apply two independent data analysis methodologies to locate stable climate states in an intermediate complexity climate model. First, drawing from the theory of quasipotentials, and viewing the state space as an energy landscape with valleys and mountain ridges, we infer the relative likelihood of the identified multistable climate states, and investigate the most likely transition trajectories as well as the expected transition times between them. Second, harnessing techniques from data science, specifically manifold learning, we characterize the data landscape of the simulation data to find climate states and basin boundaries within a fully agnostic and unsupervised framework. Both approaches show remarkable agreement, and reveal, apart from the well known warm and snowball earth states, a third intermediate stable state in one of the two climate models we consider. The combination of our approaches allows to identify how the negative feedback of ocean heat transport and entropy production via the hydrological cycle drastically change the topography of the dynamical landscape of Earth's climate.

Hierarchical nucleation in deep neural networks

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.