arXiv subjects

Pedro A. M. Mediano

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

At least 19 recordsLinked to original sources

arXiv2026

Statistical Dark Matter: Synergy is everywhere but is hard to capture

The information-theoretic construct of synergy refers to the statistical structure that is contained in three or more variables, but not in any subset of them. Here we leverage recent advances in multivariate information theory to show that synergy is far more prevalent in complex systems than previously thought. In particular, we show that many-body systems tend to become strongly dominated by synergy as the number of subcomponents grows. At the same time, our results also reveal that commonly used tools for statistical modelling severely underestimate synergistic structures. These findings imply that synergy constitutes a prevalent, yet often invisible, informational component of complex systems. With a certain poetic licence, we interpret these results as suggesting that synergy is the dark matter of statistics - interdependencies that we know exist, but standard instruments fail to detect.

cs.IT↗

arXiv2026

Hierarchy-GBP: Accelerating Factor Graph Inference via Abstraction and Recovery

Gaussian Belief Propagation (GBP) is a distributed inference algorithm that passes messages in graphical models, making it attractive for scalable spatial intelligence. However, we find GBP most effective locally: it rapidly smooths message errors that vary sharply between neighbor variables, but corrects global errors across distant graph regions incrementally through long-range message propagations. We propose Hierarchy-GBP (H-GBP), an iterative, two-stage framework that accelerates GBP by first solving these global errors with a coarse graph approximation (abstraction) and projecting the results back to the original graph (recovery), then refining the remaining local errors with GBP. We prove H-GBP convergence to the optimum by deriving the combined matrix operator of our abstraction and recovery steps and analyzing its spectral radius. Experiments on linear sparse graphs show that H-GBP converges fundamentally faster than standard GBP. Moreover, we validate H-GBP on two important spatial problems: Pose Graph Optimization (PGO) and Bundle Adjustment (BA). H-GBP markedly accelerates large-scale PGO and achieves state-of-the-art runtime across all tested BA scales.

cs.CV↗

arXiv2026

Same Compression Principle, Different Geometry: Rate-Distortion Signatures Dissociate Biological and Artificial Visual Systems

Efficient coding theory predicts that biological perceptual systems compress sensory input optimally under resource constraints, with the systematic structure of errors reflecting the geometry of that compression. Here we operationalize this principle using rate-distortion theory (RDT) to characterize how any system - biological or artificial - trades representational fidelity for informational efficiency. Treating stimulus-response behavior as an effective communication channel, we infer rate-distortion (RD) frontiers directly from confusion matrices and summarize each system with three geometric signatures: slope (beta), curvature (kappa), and area under the RD curve (AUC), capturing the marginal cost, abruptness, and overall efficiency of the accuracy-compression trade-off respectively. Applying this framework to human psychophysical data and 18 deep vision models across 12 families of controlled image perturbations at graded severities, we find that both biological and artificial systems follow a common lossy-compression principle but occupy systematically different regions of RD space. Humans exhibit smooth, flexible trade-offs characteristic of near-optimal efficient coding, while deep networks operate in steeper, more brittle regimes even at matched accuracy, with geometry dissociable from performance across training regimes. Critically, behavioral RD signatures track internal representational geometry, evidenced by the behaviorally inferred compression structure correlating with internal representational dissimilarity across all models. These results establish RD geometry as a compact diagnostic of perceptual compression strategy that recovers mechanistically interpretable structure in internal representations from behavioral input alone and extends naturally to the direct characterization of compression geometry in neural population activity.

cs.LG↗

arXiv2026

Information Dynamics of Language Communication

Quantifying how meaning propagates through communicative exchanges remains underdeveloped in computational linguistics. Here we introduce an information-theoretic framework that quantifies the directed flow of semantic content between interlocutors and decomposes multi-source contributions into redundant, unique, and synergistic components. Our approach leverages large language models as probabilistic estimators of natural language to compute two measures: semantic transfer entropy (STE), which captures directed predictive influence between speakers, and semantic partial information decomposition (SPID), which resolves how multiple sources jointly shape a target's language. Across four experiments we show that the framework detects reduced information flow in cognitively rigid dialogue, captures the dominant role of persuaders in shaping discourse, distinguishes high- from low-quality psychotherapy by the directionality of therapist-client information exchange, and reveals synergistic premise contributions in argumentative essays. This framework opens new avenues for studying information dynamics in digital discourse, pedagogical interactions, clinical dialogues, and any domain in which the structure of linguistic exchange is of research relevance.

cs.CL↗

arXiv2026

The mathematical landscape of partial information decomposition: A comprehensive review of properties and measures

Partial Information Decomposition (PID) has become one of the most prominent information-theoretic frameworks for describing the structure and quality of information in complex systems. Despite its widespread utility, there exists no unique solution constraining precisely how a PID should be constructed, leading to a multiverse of different formalisms with different mathematical commitments. In this work, we provide a comprehensive overview of the mathematical landscape of PID. By integrating existing PID measures into a common language, we systematically examine all major approaches to the PID framework that have emerged so far, determining for each measure whether or not each known property holds. In addition, we derive a web of all known theorems mapping the relationships and incompatibilities between these properties, before also revealing some novel interdependency results. In doing so, we chart a brief history of the framework, promote a unified perspective for its discussions, and offer a path towards both theoretical refinement and informed empirical applications for the future of this powerful method.

cs.IT↗

arXiv2026

Directional Confusions Reveal Divergent Inductive Biases Through Rate-Distortion Geometry in Human and Machine Vision

To humans, a robin seems more like a bird than a bird seems like a robin, but does this asymmetry also hold for machine vision? Humans and modern vision models can match each other in accuracy while making systematically different kinds of errors, differing not in how often they fail, but in who gets mistaken for whom. We show these directional confusions reveal distinct inductive biases invisible to accuracy alone. Using matched human and deep neural network responses on a natural-image categorization task under 12 perturbation types, we quantify asymmetry in confusion matrices and link its organization to the geometry of the information--error trade-off - how efficiently, and how gracefully, a system generalizes under distortion. We find that humans exhibit broad but weak asymmetries across many class pairs, whereas deep vision models show sparser, stronger directional collapses into a few dominant categories. Robustness training reduces overall asymmetry magnitude but fails to recover this human-like distributed structure. Generative simulations further show that these two asymmetry organizations shift the trade-off geometry in opposite directions even at matched accuracy, explaining why the same scalar asymmetry score can reflect fundamentally different generalization strategies. Together, these results establish directional confusion structure as a sensitive, interpretable signature of inductive bias that accuracy-based evaluation cannot recover.

cs.CV↗

arXiv2026

Integrated information theory: the good, the bad and the misunderstood

The integrated information theory of consciousness (IIT) is uniquely ambitious in proposing a mathematical formula, derived from apparently fundamental properties of conscious experience, to describe the quantity and quality of consciousness for any physical system that possesses it. IIT has generated considerable debate, which has engendered some misunderstandings and misrepresentations. Here we address and hope to remedy this. We begin by concisely summarising the essentials of IIT. Given IIT is supposed to apply universally, we do this with reference to an arbitrary patch of matter, as opposed to the usual system of discrete computational units. Then, after briefly summarising IIT's theoretical and empirical achievements, we focus on five points which we consider especially important for driving forward new theory and increasing understanding. First, a high value of the measure $Φ$ is not synonymous with `more consciousness'. We describe how $Φ$ might be replaced with a suite of quantities to obtain a multi-dimensional characterisation of states of consciousness. Second, we describe with nuance the distinct flavour of panpsychism implied by IIT -- whereby space (and time) are tiled with substrates of (proto-) consciousness -- and find this is not problematic for the theory. Third, $Φ$ is not well-defined for real physical systems, and has not been computed on any real physical system. Fourth, so far only proxies for IIT measures have been computed, and not approximations. Fifth, for IIT to fit with current successful theories in fundamental physics, a reformulation in terms of continuous fields would be needed.

q-bio.NC↗

arXiv2026

A scalable estimator of higher-order information in complex dynamical systems

Our understanding of complex systems rests on our ability to characterise how they perform distributed computation and integrate information. Advances in information theory have introduced several quantities to describe complex information structures, where collective patterns of coordination emerge from higher-order (i.e. beyond-pairwise) interdependencies. Unfortunately, the use of these approaches to study large complex systems is severely hindered by the poor scalability of existing techniques. Moreover, there are relatively few measures specifically designed for multivariate time series data. Here we introduce a novel measure of information about macroscopic structures, termed M-information, which quantifies the higher-order integration of information in complex dynamical systems. We show that M-information can be calculated via a convex optimisation problem, and we derive a robust and efficient algorithm that scales gracefully with system size. Our analyses show that M-information is resilient to noise, indexes critical behaviour in artificial neuronal populations, and reflects states of consciousness and task performance in real-world macaque and mouse neuroimaging data. Furthermore, M-information can be incorporated into existing information decomposition frameworks to reveal a comprehensive taxonomy of information dynamics. Taken together, these results help us unravel collective computation in large complex systems.

cs.IT↗

arXiv2026

From Data Statistics to Feature Geometry: How Correlations Shape Superposition

A central idea in mechanistic interpretability is that neural networks represent more features than they have dimensions, arranging them in superposition to form an over-complete basis. This framing has been influential, motivating dictionary learning approaches such as sparse autoencoders. However, superposition has mostly been studied in idealized settings where features are sparse and uncorrelated. In these settings, superposition is typically understood as introducing interference that must be minimized geometrically and filtered out by non-linearities such as ReLUs, yielding local structures like regular polytopes. We show that this account is incomplete for realistic data by introducing Bag-of-Words Superposition (BOWS), a controlled setting to encode binary bag-of-words representations of internet text in superposition. Using BOWS, we find that when features are correlated, interference can be constructive rather than just noise to be filtered out. This is achieved by arranging features according to their co-activation patterns, making interference between active features constructive, while still using ReLUs to avoid false positives. We show that this kind of arrangement is more prevalent in models trained with weight decay and naturally gives rise to semantic clusters and cyclical structures which have been observed in real language models yet were not explained by the standard picture of superposition. Code for this paper can be found at https://github.com/LucasPrietoAl/correlations-feature-geometry.

cs.LG↗

arXiv2026

Synergistic Motifs in Gaussian Systems

High-order interdependencies are central features of complex systems, yet a mechanistic explanation for their emergence remains elusive. Currently, it is unknown under what conditions high-order interdependencies, quantified by the information-theoretic construct of synergy, arise in systems governed by pairwise interactions. We solve this problem by providing precise sufficient and necessary conditions for when synergy prevails over low-order interdependencies in the weak interaction regime, namely, we prove that antibalanced (highly frustrated) correlational structures in Gaussian systems are sufficient for synergy-dominance and that antibalanced interaction motifs in Ornstein-Uhlenbeck processes are necessary for synergy-dominance. We validate the applicability of these analytical insights beyond the weak interaction regime, as well as in Ising, oscillatory, and empirical networks from multiple domains. Our results demonstrate that pairwise interactions can give rise to synergistic information in the absence of explicit high-order mechanisms, and highlight structural balance theory as an instrumental conceptual framework to study high-order interdependencies.

physics.soc-ph↗

arXiv2026

Belief Propagation Converges to Gaussian Distributions in Sparsely-Connected Factor Graphs

Belief Propagation (BP) is a powerful algorithm for distributed inference in probabilistic graphical models, however it quickly becomes infeasible for practical compute and memory budgets. Many efficient, non-parametric forms of BP have been developed, but the most popular is Gaussian Belief Propagation (GBP), a variant that assumes all distributions are locally Gaussian. GBP is widely used due to its efficiency and empirically strong performance in applications like computer vision or sensor networks - even when modelling non-Gaussian problems. In this paper, we seek to provide a theoretical guarantee for when Gaussian approximations are valid in highly non-Gaussian, sparsely-connected factor graphs performing BP (common in spatial AI). We leverage the Central Limit Theorem (CLT) to prove mathematically that variables' beliefs under BP converge to a Gaussian distribution in complex, loopy factor graphs obeying our 4 key assumptions. We then confirm experimentally that variable beliefs become increasingly Gaussian after just a few BP iterations in a stereo depth estimation task.

cs.DC↗

arXiv2026

A Brain-like Synergistic Core in LLMs Drives Behaviour and Learning

The independent evolution of intelligence in biological and artificial systems offers a unique opportunity to identify its fundamental computational principles. Here we show that large language models spontaneously develop synergistic cores -- components where information integration exceeds individual parts -- remarkably similar to those in the human brain. Using principles of information decomposition across multiple LLM model families and architectures, we find that areas in middle layers exhibit synergistic processing while early and late layers rely on redundancy, mirroring the informational organisation in biological brains. This organisation emerges through learning and is absent in randomly initialised networks. Crucially, ablating synergistic components causes disproportionate behavioural changes and performance loss, aligning with theoretical predictions about the fragility of synergy. Moreover, fine-tuning synergistic regions through reinforcement learning yields significantly greater performance gains than training redundant components, yet supervised fine-tuning shows no such advantage. This convergence suggests that synergistic information processing is a fundamental property of intelligence, providing targets for principled model design and testable predictions for biological intelligence.

cs.AI↗

arXiv2026

On the Identifiability of Regime-Switching Models with Multi-Lag Dependencies

Identifiability is central to the interpretability of deep latent variable models, ensuring parameterisations are uniquely determined by the data-generating distribution. However, it remains underexplored for deep regime-switching time series. We develop a general theoretical framework for multi-lag Regime-Switching Models (RSMs), encompassing Markov Switching Models (MSMs) and Switching Dynamical Systems (SDSs). For MSMs, we formulate the model as a temporally structured finite mixture and prove identifiability of both the number of regimes and the multi-lag transitions in a nonlinear-Gaussian setting. For SDSs, we establish identifiability of the latent variables up to permutation and scaling via temporal structure, which in turn yields conditions for identifiability of regime-dependent latent causal graphs (up to regime/node permutations). Our results hold in a fully unsupervised setting through architectural and noise assumptions that are directly enforceable via neural network design. We complement the theory with a flexible variational estimator that satisfies the assumptions and validate the results on synthetic benchmarks. Across real-world datasets from neuroscience, finance, and climate, identifiability leads to more trustworthy interpretability analysis, which is crucial for scientific discovery.

stat.ML↗

arXiv2025

Improved estimators of causal emergence for large systems

A central challenge in the study of complex systems is the quantification of emergence -- understood as the ability of the system to exhibit collective behaviours that cannot be traced down to the individual components. While recent work has proposed practical measures to detect emergence, these approaches tend to double-count the contribution of shared components, which substantially hinders their capability to effectively study large systems. In this work, we introduce a family of improved information-theoretic measures of emergence that iteratively correct for double-counted terms. Our approach is computationally efficient and provides a controllable trade-off between computational load and sensitivity, leading to more accurate and versatile estimates of emergence. The benefits of the proposed approach are demonstrated by successfully detecting emergence in both simulated and real-world data related to flocking behaviour.

physics.soc-ph↗

arXiv2025

A Logarithmic Decomposition and a Signed Measure Space for Entropy

The Shannon entropy of a random variable has much behaviour analogous to a signed measure. Previous work has explored this connection by defining a signed measure on abstract sets, which are taken to represent the information that different random variables contain. This construction is sufficient to derive many measure-theoretical counterparts to information quantities such as the mutual information (the intersection of sets), the joint entropy (the union of sets), and the conditional entropy (the difference of sets). Here we provide concrete characterisations of these abstract sets and a corresponding signed measure by extending the approach used by Yeung to all possible outcomes in an outcome space $Ω$, and in doing so 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, depending only on their structure, while also being consistent with Yeung's I-measure. We present the usability of our approach by re-examining the Gács-Körner common information and minimally sufficient statistics from this new geometric perspective and characterising it in terms of our logarithmic atoms -- a property we call logarithmic decomposability. We present possible extensions of this construction to continuous probability distributions before discussing implications for quality-led information theory. As a motivating example, we apply our new decomposition to the Dyadic and Triadic systems of James and Crutchfield and show that, in contrast to the I-measure alone, our decomposition is able to qualitatively distinguish between them.

cs.IT↗

arXiv2025

Grokking at the Edge of Numerical Stability

Grokking, the sudden generalization that occurs after prolonged overfitting, is a surprising phenomenon challenging our understanding of deep learning. Although significant progress has been made in understanding grokking, the reasons behind the delayed generalization and its dependence on regularization remain unclear. In this work, we argue that without regularization, grokking tasks push models to the edge of numerical stability, introducing floating point errors in the Softmax function, which we refer to as Softmax Collapse (SC). We demonstrate that SC prevents grokking and that mitigating SC enables grokking without regularization. Investigating the root cause of SC, we find that beyond the point of overfitting, the gradients strongly align with what we call the naïve loss minimization (NLM) direction. This component of the gradient does not alter the model's predictions but decreases the loss by scaling the logits, typically by scaling the weights along their current direction. We show that this scaling of the logits explains the delay in generalization characteristic of grokking and eventually leads to SC, halting further learning. To validate our hypotheses, we introduce two key contributions that address the challenges in grokking tasks: StableMax, a new activation function that prevents SC and enables grokking without regularization, and $\perp$Grad, a training algorithm that promotes quick generalization in grokking tasks by preventing NLM altogether. These contributions provide new insights into grokking, elucidating its delayed generalization, reliance on regularization, and the effectiveness of existing grokking-inducing methods. Code for this paper is available at https://github.com/LucasPrietoAl/grokking-at-the-edge-of-numerical-stability.

cs.LG↗

arXiv2025

Shannon invariants: A scalable approach to information decomposition

Distributed systems, such as biological and artificial neural networks, process information via complex interactions engaging multiple subsystems, resulting in high-order patterns with distinct properties across scales. Investigating how these systems process information remains challenging due to difficulties in defining appropriate multivariate metrics and ensuring their scalability to large systems. To address these challenges, we introduce a novel framework based on what we call "Shannon invariants" -- quantities that capture essential properties of high-order information processing in a way that depends only on the definition of entropy and can be efficiently calculated for large systems. Our theoretical results demonstrate how Shannon invariants can be used to resolve long-standing ambiguities regarding the interpretation of widely used multivariate information-theoretic measures. Moreover, our practical results reveal distinctive information-processing signatures of various deep learning architectures across layers, which lead to new insights into how these systems process information and how this evolves during training. Overall, our framework resolves fundamental limitations in analyzing high-order phenomena and offers broad opportunities for theoretical developments and empirical analyses.

cs.IT↗

arXiv2025

The topology of synergy: linking topological and information-theoretic approaches to higher-order interactions in complex systems

The study of irreducible higher-order interactions has become a core topic of study in complex systems. Two of the most well-developed frameworks, topological data analysis and multivariate information theory, aim to provide formal tools for identifying higher-order interactions in empirical data. Despite similar aims, however, these two approaches are built on markedly different mathematical foundations and have been developed largely in parallel. In this study, we present a head-to-head comparison of topological data analysis and information-theoretic approaches to describing higher-order interactions in multivariate data; with the aim of assessing the similarities and differences between how the frameworks define ``higher-order structures." We begin with toy examples with known topologies, before turning to naturalistic data: fMRI signals collected from the human brain. We find that intrinsic, higher-order synergistic information is associated with three-dimensional cavities in a point cloud: shapes such as spheres are synergy-dominated. In fMRI data, we find strong correlations between synergistic information and both the number and size of three-dimensional cavities. Furthermore, we find that dimensionality reduction techniques such as PCA preferentially represent higher-order redundancies, and largely fail to preserve both higher-order information and topological structure, suggesting that common manifold-based approaches to studying high-dimensional data are systematically failing to identify important features of the data. These results point towards the possibility of developing a rich theory of higher-order interactions that spans topological and information-theoretic approaches while simultaneously highlighting the profound limitations of more conventional methods.

cs.IT↗