arXiv subjects

Michael Wibral

Publications and source records attributed to Michael Wibral.

At least 19 recordsLinked to original sources

arXiv2025

Novel Inconsistency Results for Partial Information Decomposition

Partial Information Decomposition (PID) seeks to disentangle how information about a target variable is distributed across multiple sources, separating redundant, unique, and synergistic contributions. Despite extensive theoretical development and applications across diverse fields, the search for a unique, universally accepted solution remains elusive, with numerous competing proposals offering different decompositions. A promising but underutilized strategy for making progress is to establish inconsistency results, proofs that certain combinations of intuitively appealing axioms cannot be simultaneously satisfied. Such results clarify the landscape of possibilities and force us to recognize where fundamental choices must be made. In this work, we leverage the recently developed mereological approach to PID to establish novel inconsistency results with far-reaching implications. Our main theorem demonstrates that three cornerstone properties of classical information theory, namely non-negativity, the chain rule, and invariance under invertible transformations, become mutually incompatible when extended to the PID setting. This result reveals that any PID framework must sacrifice at least one property that seems fundamental to information theory itself. Additionally, we strengthen the classical result of Rauh et al., which showed that non-negativity, the identity property, and the Williams and Beer axioms cannot coexist.

cs.IT↗

arXiv2025

Redundancy Maximization as a Principle of Associative Memory Learning in Hopfield Networks

Associative memory, traditionally modeled by Hopfield networks, enables the retrieval of previously stored patterns from partial or noisy cues. Yet, the local computational principles which are required to enable this function remain incompletely understood. To formally characterize the local information processing in such systems, we employ a recent extension of information theory -- Partial Information Decomposition (PID). PID decomposes the contribution of different inputs to an output into unique information from each input, redundant information across inputs, and synergistic information that emerges from combining different inputs. Applying this framework to individual neurons in classical Hopfield networks we find that below the memory capacity, the information in a neuron's activity is characterized by high redundancy between the external pattern input and the internal recurrent input, while synergy and unique information are close to zero until the memory capacity is surpassed and performance drops steeply. Inspired by this observation, we use redundancy maximization at each neuron as an information-theoretic learning goal. This dramatically increases the network's memory capacity to 1.59, a more than tenfold improvement over the 0.14 capacity of classical Hopfield networks, and also outperforming recent state-of-the-art implementations of Hopfield networks. Overall, this work establishes redundancy maximization as a new design principle for associative memories and opens pathways for new associative memory models based on information-theoretic goals.

cs.IT↗

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↗

arXiv2024

What should a neuron aim for? Designing local objective functions based on information theory

In modern deep neural networks, the learning dynamics of the individual neurons is often obscure, as the networks are trained via global optimization. Conversely, biological systems build on self-organized, local learning, achieving robustness and efficiency with limited global information. We here show how self-organization between individual artificial neurons can be achieved by designing abstract bio-inspired local learning goals. These goals are parameterized using a recent extension of information theory, Partial Information Decomposition (PID), which decomposes the information that a set of information sources holds about an outcome into unique, redundant and synergistic contributions. Our framework enables neurons to locally shape the integration of information from various input classes, i.e. feedforward, feedback, and lateral, by selecting which of the three inputs should contribute uniquely, redundantly or synergistically to the output. This selection is expressed as a weighted sum of PID terms, which, for a given problem, can be directly derived from intuitive reasoning or via numerical optimization, offering a window into understanding task-relevant local information processing. Achieving neuron-level interpretability while enabling strong performance using local learning, our work advances a principled information-theoretic foundation for local learning strategies.

cs.IT↗

arXiv2024

Partial Information Decomposition for Continuous Variables based on Shared Exclusions: Analytical Formulation and Estimation

Describing statistical dependencies is foundational to empirical scientific research. For uncovering intricate and possibly non-linear dependencies between a single target variable and several source variables within a system, a principled and versatile framework can be found in the theory of Partial Information Decomposition (PID). Nevertheless, the majority of existing PID measures are restricted to categorical variables, while many systems of interest in science are continuous. In this paper, we present a novel analytic formulation for continuous redundancy--a generalization of mutual information--drawing inspiration from the concept of shared exclusions in probability space as in the discrete PID definition of $I^\mathrm{sx}_\cap$. Furthermore, we introduce a nearest-neighbor based estimator for continuous PID, and showcase its effectiveness by applying it to a simulated energy management system provided by the Honda Research Institute Europe GmbH. This work bridges the gap between the measure-theoretically postulated existence proofs for a continuous $I^\mathrm{sx}_\cap$ and its practical application to real-world scientific problems.

cs.IT↗

arXiv2023

From Babel to Boole: The Logical Organization of Information Decompositions

The conventional approach to the general Partial Information Decomposition (PID) problem has been redundancy-based: specifying a measure of redundant information between collections of source variables induces a PID via Moebius-Inversion over the so called redundancy lattice. Despite the prevalence of this method, there has been ongoing interest in examining the problem through the lens of different base-concepts of information, such as synergy, unique information, or union information. Yet, a comprehensive understanding of the logical organization of these different based-concepts and their associated PIDs remains elusive. In this work, we apply the mereological formulation of PID that we introduced in a recent paper to shed light on this problem. Within the mereological approach base-concepts can be expressed in terms of conditions phrased in formal logic on the specific parthood relations between the PID components and the different mutual information terms. We set forth a general pattern of these logical conditions of which all PID base-concepts in the literature are special cases and that also reveals novel base-concepts, in particular a concept we call "vulnerable information".

cs.IT↗

arXiv2023

A General Framework for Interpretable Neural Learning based on Local Information-Theoretic Goal Functions

Despite the impressive performance of biological and artificial networks, an intuitive understanding of how their local learning dynamics contribute to network-level task solutions remains a challenge to this date. Efforts to bring learning to a more local scale indeed lead to valuable insights, however, a general constructive approach to describe local learning goals that is both interpretable and adaptable across diverse tasks is still missing. We have previously formulated a local information processing goal that is highly adaptable and interpretable for a model neuron with compartmental structure. Building on recent advances in Partial Information Decomposition (PID), we here derive a corresponding parametric local learning rule, which allows us to introduce 'infomorphic' neural networks. We demonstrate the versatility of these networks to perform tasks from supervised, unsupervised and memory learning. By leveraging the interpretable nature of the PID framework, infomorphic networks represent a valuable tool to advance our understanding of the intricate structure of local learning.

cs.IT↗

arXiv2023

Information-theoretic analyses of neural data to minimize the effect of researchers' assumptions in predictive coding studies

Studies investigating neural information processing often implicitly ask both, which processing strategy out of several alternatives is used and how this strategy is implemented in neural dynamics. A prime example are studies on predictive coding. These often ask if confirmed predictions about inputs or predictions errors between internal predictions and inputs are passed on in a hierarchical neural system--while at the same time looking for the neural correlates of coding for errors and predictions. If we do not know exactly what a neural system predicts at any given moment, this results in a circular analysis--as has been criticized correctly. To circumvent such circular analysis, we propose to express information processing strategies (such as predictive coding) by local information-theoretic quantities, such that they can be estimated directly from neural data. We demonstrate our approach by investigating two opposing accounts of predictive coding-like processing strategies, where we quantify the building blocks of predictive coding, namely predictability of inputs and transfer of information, by local active information storage and local transfer entropy. We define testable hypotheses on the relationship of both quantities to identify which of the assumed strategies was used. We demonstrate our approach on spiking data from the retinogeniculate synapse of the cat. Applying our local information dynamics framework, we are able to show that the synapse codes for predictable rather than surprising input. To support our findings, we apply measures from partial information decomposition, which allow to differentiate if the transferred information is primarily bottom-up sensory input or information transferred conditionally on the current state of the synapse. Supporting our local information-theoretic results, we find that the synapse preferentially transfers bottom-up information.

q-bio.NC↗

arXiv2023

A Measure of the Complexity of Neural Representations based on Partial Information Decomposition

In neural networks, task-relevant information is represented jointly by groups of neurons. However, the specific way in which this mutual information about the classification label is distributed among the individual neurons is not well understood: While parts of it may only be obtainable from specific single neurons, other parts are carried redundantly or synergistically by multiple neurons. We show how Partial Information Decomposition (PID), a recent extension of information theory, can disentangle these different contributions. From this, we introduce the measure of "Representational Complexity", which quantifies the difficulty of accessing information spread across multiple neurons. We show how this complexity is directly computable for smaller layers. For larger layers, we propose subsampling and coarse-graining procedures and prove corresponding bounds on the latter. Empirically, for quantized deep neural networks solving the MNIST and CIFAR10 tasks, we observe that representational complexity decreases both through successive hidden layers and over training, and compare the results to related measures. Overall, we propose representational complexity as a principled and interpretable summary statistic for analyzing the structure and evolution of neural representations and complex systems in general.

cs.IT↗

arXiv2023

A Rigorous Information-Theoretic Definition of Redundancy and Relevancy in Feature Selection Based on (Partial) Information Decomposition

Selecting a minimal feature set that is maximally informative about a target variable is a central task in machine learning and statistics. Information theory provides a powerful framework for formulating feature selection algorithms -- yet, a rigorous, information-theoretic definition of feature relevancy, which accounts for feature interactions such as redundant and synergistic contributions, is still missing. We argue that this lack is inherent to classical information theory which does not provide measures to decompose the information a set of variables provides about a target into unique, redundant, and synergistic contributions. Such a decomposition has been introduced only recently by the partial information decomposition (PID) framework. Using PID, we clarify why feature selection is a conceptually difficult problem when approached using information theory and provide a novel definition of feature relevancy and redundancy in PID terms. From this definition, we show that the conditional mutual information (CMI) maximizes relevancy while minimizing redundancy and propose an iterative, CMI-based algorithm for practical feature selection. We demonstrate the power of our CMI-based algorithm in comparison to the unconditional mutual information on benchmark examples and provide corresponding PID estimates to highlight how PID allows to quantify information contribution of features and their interactions in feature-selection problems.

cs.IT↗

arXiv2022

Intrinsic timescales of spiking activity in humans during wakefulness and sleep

Information processing in the brain requires integration of information over time. Such an integration can be achieved if signals are maintained in the network activity for the required period, as quantified by the intrinsic timescale. While short timescales are considered beneficial for fast responses to stimuli, long timescales facilitate information storage and integration. We quantified intrinsic timescales from spiking activity in the medial temporal lobe of humans. We found extended and highly diverse timescales ranging from tens to hundreds of milliseconds, though with no evidence for differences between subareas. Notably, however, timescales differed between sleep stages and were longest during slow wave sleep. This supports the hypothesis that intrinsic timescales are a central mechanism to tune networks to the requirements of different tasks and cognitive states.

q-bio.NC↗

arXiv2022

Dendritic predictive coding: A theory of cortical computation with spiking neurons

Top-down feedback in cortex is critical for guiding sensory processing, which has prominently been formalized in the theory of hierarchical predictive coding (hPC). However, experimental evidence for error units, which are central to the theory, is inconclusive, and it remains unclear how hPC can be implemented with spiking neurons. To address this, we connect hPC to existing work on efficient coding in balanced networks with lateral inhibition, and predictive computation at apical dendrites. Together, this work points to an efficient implementation of hPC with spiking neurons, where prediction errors are computed not in separate units, but locally in dendritic compartments. The implied model shows a remarkable correspondence to experimentally observed cortical connectivity patterns, plasticity and dynamics, and at the same time can explain hallmarks of predictive processing, such as mismatch responses, in cortex. We thus propose dendritic predictive coding as one of the main organizational principles of cortex.

q-bio.NC↗

arXiv2022

Bits and Pieces: Understanding Information Decomposition from Part-whole Relationships and Formal Logic

Partial information decomposition (PID) seeks to decompose the multivariate mutual information that a set of source variables contains about a target variable into basic pieces, the so called "atoms of information". Each atom describes a distinct way in which the sources may contain information about the target. In this paper we show, first, that the entire theory of partial information decomposition can be derived from considerations of elementary parthood relationships between information contributions. This way of approaching the problem has the advantage of directly characterizing the atoms of information, instead of taking an indirect approach via the concept of redundancy. Secondly, we describe several intriguing links between PID and formal logic. In particular, we show how to define a measure of PID based on the information provided by certain statements about source realizations. Furthermore, we show how the mathematical lattice structure underlying PID theory can be translated into an isomorphic structure of logical statements with a particularly simple ordering relation: logical implication. The conclusion to be drawn from these considerations is that there are three isomorphic "worlds" of partial information decomposition, i.e. three equivalent ways to mathematically describe the decomposition of the information carried by a set of sources about a target: the world of parthood relationships, the world of logical statements, and the world of antichains that was utilized by Williams and Beer in their original exposition of PID theory. We additionally show how the parthood perspective provides a systematic way to answer a type of question that has been much discussed in the PID field: whether a partial information decomposition can be uniquely determined based on concepts other than redundant information.

cs.AI↗

arXiv2021

Estimating the Unique Information of Continuous Variables

The integration and transfer of information from multiple sources to multiple targets is a core motive of neural systems. The emerging field of partial information decomposition (PID) provides a novel information-theoretic lens into these mechanisms by identifying synergistic, redundant, and unique contributions to the mutual information between one and several variables. While many works have studied aspects of PID for Gaussian and discrete distributions, the case of general continuous distributions is still uncharted territory. In this work we present a method for estimating the unique information in continuous distributions, for the case of one versus two variables. Our method solves the associated optimization problem over the space of distributions with fixed bivariate marginals by combining copula decompositions and techniques developed to optimize variational autoencoders. We obtain excellent agreement with known analytic results for Gaussians, and illustrate the power of our new approach in several brain-inspired neural models. Our method is capable of recovering the effective connectivity of a chaotic network of rate neurons, and uncovers a complex trade-off between redundancy, synergy and unique information in recurrent networks trained to solve a generalized XOR task.

cs.IT↗

arXiv2021

A partial information decomposition for discrete and continuous variables

Conceptually, partial information decomposition (PID) is concerned with separating the information contributions several sources hold about a certain target by decomposing the corresponding joint mutual information into contributions such as synergistic, redundant, or unique information. Despite PID conceptually being defined for any type of random variables, so far, PID could only be quantified for the joint mutual information of discrete systems. Recently, a quantification for PID in continuous settings for two or three source variables was introduced. Nonetheless, no ansatz has managed to both quantify PID for more than three variables and cover general measure-theoretic random variables, such as mixed discrete-continuous, or continuous random variables yet. In this work we will propose an information quantity, defining the terms of a PID, which is well-defined for any number or type of source or target random variable. This proposed quantity is tightly related to a recently developed local shared information quantity for discrete random variables based on the idea of shared exclusions. Further, we prove that this newly proposed information-measure fulfills various desirable properties, such as satisfying a set of local PID axioms, invariance under invertible transformations, differentiability with respect to the underlying probability density, and admitting a target chain rule.

cs.IT↗

arXiv2021

Introducing a differentiable measure of pointwise shared information

Partial information decomposition (PID) of the multivariate mutual information describes the distinct ways in which a set of source variables contains information about a target variable. The groundbreaking work of Williams and Beer has shown that this decomposition cannot be determined from classic information theory without making additional assumptions, and several candidate measures have been proposed, often drawing on principles from related fields such as decision theory. None of these measures is differentiable with respect to the underlying probability mass function. We here present a novel measure that satisfies this property, emerges solely from information-theoretic principles, and has the form of a local mutual information. We show how the measure can be understood from the perspective of exclusions of probability mass, a principle that is foundational to the original definition of the mutual information by Fano. Since our measure is well-defined for individual realizations of the random variables it lends itself for example to local learning in artificial neural networks. We also show that it has a meaningful Möbius inversion on a redundancy lattice and obeys a target chain rule. We give an operational interpretation of the measure based on the decisions that an agent should take if given only the shared information.

cs.IT↗

arXiv2020

The challenges of containing SARS-CoV-2 via test-trace-and-isolate

Without a cure, vaccine, or proven long-term immunity against SARS-CoV-2, test-trace-and-isolate (TTI) strategies present a promising tool to contain its spread. For any TTI strategy, however, mitigation is challenged by pre- and asymptomatic transmission, TTI-avoiders, and undetected spreaders, who strongly contribute to hidden infection chains. Here, we studied a semi-analytical model and identified two tipping points between controlled and uncontrolled spread: (1) the behavior-driven reproduction number of the hidden chains becomes too large to be compensated by the TTI capabilities, and (2) the number of new infections exceeds the tracing capacity. Both trigger a self-accelerating spread. We investigated how these tipping points depend on challenges like limited cooperation, missing contacts, and imperfect isolation. Our model results suggest that TTI alone is insufficient to contain an otherwise unhindered spread of SARS-CoV-2, implying that complementary measures like social distancing and improved hygiene remain necessary.

q-bio.PE↗

arXiv2020

Inferring change points in the COVID-19 spreading reveals the effectiveness of interventions

As COVID-19 is rapidly spreading across the globe, short-term modeling forecasts provide time-critical information for decisions on containment and mitigation strategies. A main challenge for short-term forecasts is the assessment of key epidemiological parameters and how they change when first interventions show an effect. By combining an established epidemiological model with Bayesian inference, we analyze the time dependence of the effective growth rate of new infections. Focusing on the COVID-19 spread in Germany, we detect change points in the effective growth rate that correlate well with the times of publicly announced interventions. Thereby, we can quantify the effect of interventions, and we can incorporate the corresponding change points into forecasts of future scenarios and case numbers. Our code is freely available and can be readily adapted to any country or region.

q-bio.PE↗