arXiv subjects

Pedro A. M. Mediano

Publications and source records attributed to Pedro A. M. Mediano.

At least 37 records · Page 2Linked to original sources

arXiv2025

Simple physical systems as a reference for multivariate information dynamics

Understanding a complex system entails capturing the non-trivial collective phenomena that arise from interactions between its different parts. Information theory is a flexible and robust framework to study such behaviours, with several measures designed to quantify and characterise the interdependencies among the system's components. However, since these estimators rely on the statistical distributions of observed quantities, it is crucial to examine the relationships between information-theoretic measures and the system's underlying mechanistic structure. To this end, here we present an information-theoretic analytical investigation of an elementary system of interactive random walkers subject to Gaussian noise. Focusing on partial information decomposition, causal emergence, and integrated information, our results help us develop some intuitions on their relationship with the physical parameters of the system. For instance, we observe that uncoupled systems can exhibit emergent properties, in a way that we suggest may be better described as ''statistically autonomous''. Overall, we observe that in this simple scenario information measures align more reliably with the system's mechanistic properties when calculated at the level of microscopic components, rather than their coarse-grained counterparts, and over timescales comparable with the system's intrinsic dynamics. Moreover, we show that approaches that separate the contributions of the system's dynamics and steady-state distribution (e.g. via causal perturbations) may help strengthen the interpretation of information-theoretic analyses.

cs.IT↗

arXiv2025

From Lazy to Rich: Exact Learning Dynamics in Deep Linear Networks

Biological and artificial neural networks develop internal representations that enable them to perform complex tasks. In artificial networks, the effectiveness of these models relies on their ability to build task specific representation, a process influenced by interactions among datasets, architectures, initialization strategies, and optimization algorithms. Prior studies highlight that different initializations can place networks in either a lazy regime, where representations remain static, or a rich/feature learning regime, where representations evolve dynamically. Here, we examine how initialization influences learning dynamics in deep linear neural networks, deriving exact solutions for lambda-balanced initializations-defined by the relative scale of weights across layers. These solutions capture the evolution of representations and the Neural Tangent Kernel across the spectrum from the rich to the lazy regimes. Our findings deepen the theoretical understanding of the impact of weight initialization on learning regimes, with implications for continual learning, reversal learning, and transfer learning, relevant to both neuroscience and practical applications.

cs.LG↗

arXiv2024

Null models for comparing information decomposition across complex systems

A key feature of information theory is its universality, as it can be applied to study a broad variety of complex systems. However, many information-theoretic measures can vary significantly even across systems with similar properties, making normalisation techniques essential for allowing meaningful comparisons across datasets. Inspired by the framework of Partial Information Decomposition (PID), here we introduce Null Models for Information Theory (NuMIT), a null model-based non-linear normalisation procedure which improves upon standard entropy-based normalisation approaches and overcomes their limitations. We provide practical implementations of the technique for systems with different statistics, and showcase the method on synthetic models and on human neuroimaging data. Our results demonstrate that NuMIT provides a robust and reliable tool to characterise complex systems of interest, allowing cross-dataset comparisons and providing a meaningful significance test for PID analyses.

cs.IT↗

arXiv2024

Characterising high-order interdependence via entropic conjugation

High-order phenomena play crucial roles in many systems of interest, but their analysis is often highly nontrivial. There is a rich literature providing a number of alternative information-theoretic quantities capturing high-order phenomena, but their interpretation and relationship with each other is not well understood. The lack of principles unifying these quantities obscures the choice of tools for enabling specific type of analyses. Here we show how an entropic conjugation provides a theoretically grounded principle to investigate the space of possible high-order quantities, clarifying the nature of the existent metrics while revealing gaps in the literature. This leads to identify novel notions of symmetry and skew-symmetry as key properties for guaranteeing a balanced account of high-order interdependencies and enabling broadly applicable analyses across physical systems.

cs.IT↗

arXiv2024

The Fast Möbius Transform: An algebraic approach to information decomposition

The partial information decomposition (PID) and its extension integrated information decomposition ($Φ$ID) are promising frameworks to investigate information phenomena involving multiple variables. An important limitation of these approaches is the high computational cost involved in their calculation. Here we leverage fundamental algebraic properties of these decompositions to enable a computationally-efficient method to estimate them, which we call the fast Möbius transform. Our approach is based on a novel formula for estimating the Möbius function that circumvents important computational bottlenecks. We showcase the capabilities of this approach by presenting two analyses that would be unfeasible without this method: decomposing the information that neural activity at different frequency bands yield about the brain's macroscopic functional organisation, and identifying distinctive dynamical properties of the interactions between multiple voices in baroque music. Overall, our proposed approach illuminates the value of algebraic facets of information decomposition and opens the way to a wide range of future analyses.

cs.IT↗

arXiv2024

Algebraic Representations of Entropy and Fixed-Parity Information Quantities

Many information-theoretic quantities have corresponding representations in terms of sets. The prevailing signed measure space for characterising entropy, the $I$-measure of Yeung, is occasionally unable to discern between qualitatively distinct systems. In previous work, we presented a refinement of this signed measure space and demonstrated its capability to represent many quantities, which we called logarithmically decomposable quantities. In the present work we demonstrate that this framework has natural algebraic behaviour which can be expressed in terms of ideals (characterised here as upper-sets), and we show that this behaviour allows us to make various counting arguments and characterise many fixed-parity information quantity expressions. As an application, we give an algebraic proof that the only completely synergistic system of three finite variables $X$, $Y$ and $Z = f(X,Y)$ is the XOR gate.

cs.IT↗

arXiv2024

Evolving reservoir computers reveals bidirectional coupling between predictive power and emergent dynamics

Biological neural networks can perform complex computations to predict their environment, far above the limited predictive capabilities of individual neurons. While conventional approaches to understanding these computations often focus on isolating the contributions of single neurons, here we argue that a deeper understanding requires considering emergent dynamics - dynamics that make the whole system "more than the sum of its parts". Specifically, we examine the relationship between prediction performance and emergence by leveraging recent quantitative metrics of emergence, derived from Partial Information Decomposition, and by modelling the prediction of environmental dynamics in a bio-inspired computational framework known as reservoir computing. Notably, we reveal a bidirectional coupling between prediction performance and emergence, which generalises across task environments and reservoir network topologies, and is recapitulated by three key results: 1) Optimising hyperparameters for performance enhances emergent dynamics, and vice versa; 2) Emergent dynamics represent a near sufficient criterion for prediction success in all task environments, and an almost necessary criterion in most environments; 3) Training reservoir computers on larger datasets results in stronger emergent dynamics, which contain task-relevant information crucial for performance. Overall, our study points to a pivotal role of emergence in facilitating environmental predictions in a bio-inspired computational architecture.

q-bio.NC↗

arXiv2024

Identifying Nonstationary Causal Structures with High-Order Markov Switching Models

Causal discovery in time series is a rapidly evolving field with a wide variety of applications in other areas such as climate science and neuroscience. Traditional approaches assume a stationary causal graph, which can be adapted to nonstationary time series with time-dependent effects or heterogeneous noise. In this work we address nonstationarity via regime-dependent causal structures. We first establish identifiability for high-order Markov Switching Models, which provide the foundations for identifiable regime-dependent causal discovery. Our empirical studies demonstrate the scalability of our proposed approach for high-order regime-dependent structure estimation, and we illustrate its applicability on brain activity data.

stat.ML↗

arXiv2024

Software in the natural world: A computational approach to hierarchical emergence

Understanding the functional architecture of complex systems is crucial to illuminate their inner workings and enable effective methods for their prediction and control. Recent advances have introduced tools to characterise emergent macroscopic levels; however, while these approaches are successful in identifying when emergence takes place, they are limited in the extent they can determine how it does. Here we address this limitation by developing a computational approach to emergence, which characterises macroscopic processes in terms of their computational capabilities. Concretely, we articulate a view on emergence based on how software works, which is rooted on a mathematical formalism that articulates how macroscopic processes can express self-contained informational, interventional, and computational properties. This framework establishes a hierarchy of nested self-contained processes that determines what computations take place at what level, which in turn delineates the functional architecture of a complex system. This approach is illustrated on paradigmatic models from the statistical physics and computational neuroscience literature, which are shown to exhibit macroscopic processes that are akin to software in human-engineered systems. Overall, this framework enables a deeper understanding of the multi-level structure of complex systems, revealing specific ways in which they can be efficiently simulated, predicted, and controlled.

nlin.AO↗

arXiv2024

Characterising directed and undirected metrics of high-order interdependence

Systems of interest for theoretical or experimental work often exhibit high-order interactions, corresponding to statistical interdependencies in groups of variables that cannot be reduced to dependencies in subsets of them. While still under active development, the framework of partial information decomposition (PID) has emerged as the dominant approach to conceptualise and calculate high-order interdependencies. PID approaches can be grouped in two types: directed approaches that divide variables into sources and targets, and undirected approaches that treat all variables equally. Directed and undirected approaches are usually employed to investigate different scenarios, and hence little is known about how these two types of approaches may relate to each other, or if their corresponding quantities are linked in some way. In this paper we investigate the relationship between the redundancy-synergy index (RSI) and the O-information, which are practical metrics of directed and undirected high-order interdependencies, respectively. Our results reveal tight links between these two quantities, and provide interpretations of them in terms of likelihood ratios in a hypothesis testing setting, as well as in terms of projections in information geometry.

cs.IT↗

arXiv2023

Interaction Measures, Partition Lattices and Kernel Tests for High-Order Interactions

Models that rely solely on pairwise relationships often fail to capture the complete statistical structure of the complex multivariate data found in diverse domains, such as socio-economic, ecological, or biomedical systems. Non-trivial dependencies between groups of more than two variables can play a significant role in the analysis and modelling of such systems, yet extracting such high-order interactions from data remains challenging. Here, we introduce a hierarchy of $d$-order ($d \geq 2$) interaction measures, increasingly inclusive of possible factorisations of the joint probability distribution, and define non-parametric, kernel-based tests to establish systematically the statistical significance of $d$-order interactions. We also establish mathematical links with lattice theory, which elucidate the derivation of the interaction measures and their composite permutation tests; clarify the connection of simplicial complexes with kernel matrix centring; and provide a means to enhance computational efficiency. We illustrate our results numerically with validations on synthetic data, and through an application to neuroimaging data.

stat.ML↗

arXiv2023

Quantifying Hierarchical Selection

At what level does selective pressure effectively act? When considering the reproductive dynamics of interacting and mutating agents, it has long been debated whether selection is better understood by focusing on the individual or if hierarchical selection emerges as a consequence of joint adaptation. Despite longstanding efforts in theoretical ecology there is still no consensus on this fundamental issue, most likely due to the difficulty in obtaining adequate data spanning sufficient number of generations and the lack of adequate tools to quantify the effect of hierarchical selection. Here we capitalise on recent advances in information-theoretic data analysis to advance this state of affairs by investigating the emergence of high-order structures -- such as groups of species -- in the collective dynamics of the Tangled Nature model of evolutionary ecology. Our results show that evolutionary dynamics can lead to clusters of species that act as a selective group, that acquire information-theoretic agency. Overall, our findings provide quantitative evidence supporting the relevance of high-order structures in evolutionary ecology, which can emerge even from relatively simple processes of adaptation and selection.

q-bio.PE↗

arXiv2023

Information decomposition reveals hidden high-order contributions to temporal irreversibility

Temporal irreversibility, often referred to as the arrow of time, is a fundamental concept in statistical mechanics. Markers of irreversibility also provide a powerful characterisation of information processing in biological systems. However, current approaches tend to describe temporal irreversibility in terms of a single scalar quantity, without disentangling the underlying dynamics that contribute to irreversibility. Here we propose a broadly applicable information-theoretic framework to characterise the arrow of time in multivariate time series, which yields qualitatively different types of irreversible information dynamics. This multidimensional characterisation reveals previously unreported high-order modes of irreversibility, and establishes a formal connection between recent heuristic markers of temporal irreversibility and metrics of information processing. We demonstrate the prevalence of high-order irreversibility in the hyperactive regime of a biophysical model of brain dynamics, showing that our framework is both theoretically principled and empirically useful. This work challenges the view of the arrow of time as a monolithic entity, enhancing both our theoretical understanding of irreversibility and our ability to detect it in practical applications.

q-bio.NC↗

arXiv2023

Quantifying synergy and redundancy in multiplex networks

Understanding how different networks relate to each other is key for obtaining a greater insight into complex systems. Here, we introduce an intuitive yet powerful framework to characterise the relationship between two networks comprising the same nodes. We showcase our framework by decomposing the shortest paths between nodes as being contributed uniquely by one or the other source network, or redundantly by either, or synergistically by the two together. Our approach takes into account the networks' full topology, and it also provides insights at multiple levels of resolution: from global statistics, to individual paths of different length. We show that this approach is widely applicable, from brains to the London public transport system. In humans and across 123 other mammalian species, we demonstrate that reliance on unique contributions by long-range white matter fibers is a conserved feature of mammalian structural brain networks. Across species, we also find that efficient communication relies on significantly greater synergy between long-range and short-range fibers than expected by chance, and significantly less redundancy. Our framework may find applications to help decide how to trade-off different desiderata when designing network systems, or to evaluate their relative presence in existing systems, whether biological or artificial.

cs.SI↗

arXiv2023

Dynamical noise can enhance high-order statistical structure in complex systems

Recent research has provided a wealth of evidence highlighting the pivotal role of high-order interdependencies in supporting the information-processing capabilities of distributed complex systems. These findings may suggest that high-order interdependencies constitute a powerful resource that is, however, challenging to harness and can be readily disrupted. In this paper we contest this perspective by demonstrating that high-order interdependencies can not only exhibit robustness to stochastic perturbations, but can in fact be enhanced by them. Using elementary cellular automata as a general testbed, our results unveil the capacity of dynamical noise to enhance the statistical regularities between agents and, intriguingly, even alter the prevailing character of their interdependencies. Furthermore, our results show that these effects are related to the high-order structure of the local rules, which affect the system's susceptibility to noise and characteristic times-scales. These results deepen our understanding of how high-order interdependencies may spontaneously emerge within distributed systems interacting with stochastic environments, thus providing an initial step towards elucidating their origin and function in complex systems like the human brain.

nlin.AO↗

arXiv2023

A Logarithmic Decomposition for Information

The Shannon entropy of a random variable $X$ has much behaviour analogous to a signed measure. Previous work has concretized this connection by defining a signed measure $μ$ on an abstract information space $\tilde{X}$, which is taken to represent the information that $X$ contains. This construction is sufficient to derive many measure-theoretical counterparts to information quantities such as the mutual information $I(X; Y) = μ(\tilde{X} \cap \tilde{Y})$, the joint entropy $H(X,Y) = μ(\tilde{X} \cup \tilde{Y})$, and the conditional entropy $H(X|Y) = μ(\tilde{X}\, \setminus \, \tilde{Y})$. We demonstrate that there exists a much finer decomposition with intuitive properties which we call the logarithmic decomposition (LD). We show that this signed measure space has the useful property that its logarithmic atoms are easily characterised with negative or positive entropy, while also being coherent with Yeung's $I$-measure. We present the usability of our approach by re-examining the Gács-Körner common information from this new geometric perspective and characterising it in terms of our logarithmic atoms. We then highlight that our geometric refinement can account for an entire class of information quantities, which we call logarithmically decomposable quantities.

cs.IT↗

arXiv2023

Bayesian at heart: Towards autonomic outflow estimation via generative state-space modelling of heart rate dynamics

Recent research is revealing how cognitive processes are supported by a complex interplay between the brain and the rest of the body, which can be investigated by the analysis of physiological features such as breathing rhythms, heart rate, and skin conductance. Heart rate dynamics are of particular interest as they provide a way to track the sympathetic and parasympathetic outflow from the autonomic nervous system, which is known to play a key role in modulating attention, memory, decision-making, and emotional processing. However, extracting useful information from heartbeats about the autonomic outflow is still challenging due to the noisy estimates that result from standard signal-processing methods. To advance this state of affairs, we propose a paradigm shift in how we conceptualise and model heart rate: instead of being a mere summary of the observed inter-beat intervals, we introduce a modelling framework that views heart rate as a hidden stochastic process that drives the observed heartbeats. Moreover, by leveraging the rich literature of state-space modelling and Bayesian inference, our proposed framework delivers a description of heart rate dynamics that is not a point estimate but a posterior distribution of a generative model. We illustrate the capabilities of our method by showing that it recapitulates linear properties of conventional heart rate estimators, while exhibiting a better discriminative power for metrics of dynamical complexity compared across different physiological states.

q-bio.QM↗

arXiv2022

Synergistic information supports modality integration and flexible learning in neural networks solving multiple tasks

Striking progress has recently been made in understanding human cognition by analyzing how its neuronal underpinnings are engaged in different modes of information processing. Specifically, neural information can be decomposed into synergistic, redundant, and unique features, with synergistic components being particularly aligned with complex cognition. However, two fundamental questions remain unanswered: (a) precisely how and why a cognitive system can become highly synergistic; and (b) how these informational states map onto artificial neural networks in various learning modes. To address these questions, here we employ an information-decomposition framework to investigate the information processing strategies adopted by simple artificial neural networks performing a variety of cognitive tasks in both supervised and reinforcement learning settings. Our results show that synergy increases as neural networks learn multiple diverse tasks. Furthermore, performance in tasks requiring integration of multiple information sources critically relies on synergistic neurons. Finally, randomly turning off neurons during training through dropout increases network redundancy, corresponding to an increase in robustness. Overall, our results suggest that while redundant information is required for robustness to perturbations in the learning process, synergistic information is used to combine information from multiple modalities -- and more generally for flexible and efficient learning. These findings open the door to new ways of investigating how and why learning systems employ specific information-processing strategies, and support the principle that the capacity for general-purpose learning critically relies in the system's information dynamics.

q-bio.NC↗