arXiv subjects

David Dahmen

Publications and source records attributed to David Dahmen.

At least 19 recordsLinked to original sources

arXiv2025

Reduction of interaction order in hard combinatorial optimization via conditionally independent degrees of freedom

Combinatorial optimization problems have a broad range of applications and map to physical systems with complex dynamics. Among them, the 3-SAT problem is prominent due to its NP-complete nature. In physics terms, its solution corresponds to finding the ground state of a disordered Ising spin Hamiltonian with third-order, or tensor, interactions. The large growth of the number of third-order interactions with number of variables poses technical difficulties for the physical implementation of minimizers. Therefore, researchers have proposed quadratization techniques which reduce the order of the system, however, at the cost of including additional degrees of freedom. Their inclusion induces a drastic slow down in the minimization, which makes such procedures technically infeasible for large problems. In this work, we take a physics approach by employing the renormalization group to create a pairwise interacting system from the original third-order system while preserving the free energy. Our procedure utilizes additional degrees of freedom that exhibit an independent dynamics provided the original degrees of freedom are fixed. A step-wise trace of the extra variables while running the minimization is therefore theoretically manageable, yielding a state-dependent effective interaction. We use the effective interaction to reconstruct the original third-order energy spectrum, as this yields equal scaling of computations-to-ground-state compared to the original tensor formulation. Here, the original degrees of freedom interact with a subsystem that appears to be in a superposition of an exponentially large number of states. In the zero-temperature limit, the superposition concentrates on one state. Our spectrum-engineering techniques reveal new routes toward the ground state of disordered Ising systems, through Markov chains, and allow for efficient technological implementations.

cond-mat.dis-nn↗

arXiv2025

Event-driven eligibility propagation in large sparse networks: efficiency shaped by biological realism

Despite remarkable technological advances, AI systems may still benefit from biological principles, such as recurrent connectivity and energy-efficient mechanisms. Drawing inspiration from the brain, we present a biologically plausible extension of the eligibility propagation (e-prop) learning rule for recurrent spiking networks. By translating the time-driven update scheme into an event-driven one, we integrate the learning rule into a simulation platform for large-scale spiking neural networks and demonstrate its applicability to tasks such as neuromorphic MNIST. We extend the model with prominent biological features such as continuous dynamics and weight updates, strict locality, and sparse connectivity. Our results show that biologically grounded constraints can inform the design of computationally efficient AI algorithms, offering scalability to millions of neurons without compromising learning performance. This work bridges machine learning and computational neuroscience, paving the way for sustainable, biologically inspired AI systems while advancing our understanding of brain-like learning.

cs.NE↗

arXiv2025

Beyond-mean-field fluctuations for the solution of constraint satisfaction problems

Constraint Satisfaction Problems (CSPs) lie at the heart of complexity theory and find application in a plethora of prominent tasks ranging from cryptography to genetics. Classical approaches use Hopfield networks to find approximate solutions while recently, modern machine-learning techniques like graph neural networks have become popular for this task. In this study, we employ the known mapping of MAX-2-SAT, a class of CSPs, to a spin-glass system from statistical physics, and use Glauber dynamics to approximately find its ground state, which corresponds to the optimal solution of the underlying problem. We show that Glauber dynamics outperforms the traditional Hopfield-network approach and can compete with state-of-the-art solvers. A systematic theoretical analysis uncovers the role of stochastic fluctuations in finding CSP solutions: even in the absence of thermal fluctuations at $T=0$ a significant portion of spins, which correspond to the CSP variables, attains an effective spin-dependent non-zero temperature. These spins form a subspace in which the stochastic Glauber dynamics continuously performs flips to eventually find better solutions. This is possible since the energy is degenerate, such that spin flips in this free-spin space do not require energy. Our theoretical analysis leads to deterministic solvers that effectively account for such fluctuations, thereby reaching state-of-the-art performance.

cond-mat.dis-nn↗

arXiv2025

From Kernels to Features: A Multi-Scale Adaptive Theory of Feature Learning

Feature learning in neural networks is crucial for their expressive power and inductive biases, motivating various theoretical approaches. Some approaches describe network behavior after training through a change in kernel scale from initialization, resulting in a generalization power comparable to a Gaussian process. Conversely, in other approaches training results in the adaptation of the kernel to the data, involving directional changes to the kernel. The relationship and respective strengths of these two views have so far remained unresolved. This work presents a theoretical framework of multi-scale adaptive feature learning bridging these two views. Using methods from statistical mechanics, we derive analytical expressions for network output statistics which are valid across scaling regimes and in the continuum between them. A systematic expansion of the network's probability distribution reveals that mean-field scaling requires only a saddle-point approximation, while standard scaling necessitates additional correction terms. Remarkably, we find across regimes that kernel adaptation can be reduced to an effective kernel rescaling when predicting the mean network output in the special case of a linear network. However, for linear and non-linear networks, the multi-scale adaptive approach captures directional feature learning effects, providing richer insights than what could be recovered from a rescaling of the kernel alone.

cond-mat.dis-nn↗

arXiv2024

Identifying the impact of local connectivity patterns on dynamics in excitatory-inhibitory networks

Networks of excitatory and inhibitory (EI) neurons form a canonical circuit in the brain. Seminal theoretical results on dynamics of such networks are based on the assumption that synaptic strengths depend on the type of neurons they connect, but are otherwise statistically independent. Recent synaptic physiology datasets however highlight the prominence of specific connectivity patterns that go well beyond what is expected from independent connections. While decades of influential research have demonstrated the strong role of the basic EI cell type structure, to which extent additional connectivity features influence dynamics remains to be fully determined. Here we examine the effects of pairwise connectivity motifs on the linear dynamics in EI networks using an analytical framework that approximates the connectivity in terms of low-rank structures. This low-rank approximation is based on a mathematical derivation of the dominant eigenvalues of the connectivity matrix and predicts the impact on responses to external inputs of connectivity motifs and their interactions with cell-type structure. Our results reveal that a particular pattern of connectivity, chain motifs, have a much stronger impact on dominant eigenmodes than other pairwise motifs. An overrepresentation of chain motifs induces a strong positive eigenvalue in inhibition-dominated networks and generates a potential instability that requires revisiting the classical excitation-inhibition balance criteria. Examining effects of external inputs, we show that chain motifs can on their own induce paradoxical responses where an increased input to inhibitory neurons leads to a decrease in their activity due to the recurrent feedback. These findings have direct implications for the interpretation of experiments in which responses to optogenetic perturbations are measured and used to infer the dynamical regime of cortical circuits.

q-bio.NC↗

arXiv2024

Critical feature learning in deep neural networks

A key property of neural networks driving their success is their ability to learn features from data. Understanding feature learning from a theoretical viewpoint is an emerging field with many open questions. In this work we capture finite-width effects with a systematic theory of network kernels in deep non-linear neural networks. We show that the Bayesian prior of the network can be written in closed form as a superposition of Gaussian processes, whose kernels are distributed with a variance that depends inversely on the network width N . A large deviation approach, which is exact in the proportional limit for the number of data points $P = αN \rightarrow \infty$, yields a pair of forward-backward equations for the maximum a posteriori kernels in all layers at once. We study their solutions perturbatively to demonstrate how the backward propagation across layers aligns kernels with the target. An alternative field-theoretic formulation shows that kernel adaptation of the Bayesian posterior at finite-width results from fluctuations in the prior: larger fluctuations correspond to a more flexible network prior and thus enable stronger adaptation to data. We thus find a bridge between the classical edge-of-chaos NNGP theory and feature learning, exposing an intricate interplay between criticality, response functions, and feature scale.

cond-mat.dis-nn↗

arXiv2024

Effect of Synaptic Heterogeneity on Neuronal Coordination

Recent advancements in measurement techniques have resulted in an increasing amount of data on neural activities recorded in parallel, revealing largely heterogeneous correlation patterns across neurons. Yet, the mechanistic origin of this heterogeneity is largely unknown because existing theoretical approaches linking structure and dynamics in neural circuits are restricted to population-averaged connectivity and activity. Here we present a systematic inclusion of heterogeneity in network connectivity to derive quantitative predictions for neuron-resolved covariances and their statistics in spiking neural networks. Our study shows that the heterogeneity in covariances is not a result of variability in single-neuron firing statistics but stems from the ubiquitously observed sparsity and variability of connections in brain networks. Linear-response theory maps these features to the effective connectivity between neurons, which in turn determines neuronal covariances. Beyond-mean-field tools reveal that synaptic heterogeneity modulates the variability of covariances and thus the complexity of neuronal coordination across many orders of magnitude.

cond-mat.dis-nn↗

arXiv2023

A theory of data variability in Neural Network Bayesian inference

Bayesian inference and kernel methods are well established in machine learning. The neural network Gaussian process in particular provides a concept to investigate neural networks in the limit of infinitely wide hidden layers by using kernel and inference methods. Here we build upon this limit and provide a field-theoretic formalism which covers the generalization properties of infinitely wide networks. We systematically compute generalization properties of linear, non-linear, and deep non-linear networks for kernel matrices with heterogeneous entries. In contrast to currently employed spectral methods we derive the generalization properties from the statistical properties of the input, elucidating the interplay of input dimensionality, size of the training data set, and variability of the data. We show that data variability leads to a non-Gaussian action reminiscent of a ($φ^3+φ^4$)-theory. Using our formalism on a synthetic task and on MNIST we obtain a homogeneous kernel matrix approximation for the learning curve as well as corrections due to data variability which allow the estimation of the generalization properties and exact results for the bounds of the learning curves in the case of infinitely many training data points.

cond-mat.dis-nn↗

arXiv2023

Hidden connectivity structures control collective network dynamics

Many observables of brain dynamics appear to be optimized for computation. Which connectivity structures underlie this fine-tuning? We propose that many of these structures are naturally encoded in the space that more directly relates to network dynamics - the space of the connectivity eigenmodes. We develop a mathematical theory to impose eigenmode structures on connectivity, systematically characterizing their effect on network dynamics. We find the density of nearly-critical eigenvalues to be a particularly fundamental structure. It flexibly controls the power-law scaling of dynamical observables, in analogy with the system's spatial dimension in classical critical phenomena. This mechanism provides control over observables which are found to be fine-tuned in brain networks, but remained so far unexplained by traditionally studied structures, such as connectivity motifs. Specifically, the slope of the principal component spectrum of neural activity can be fine-tuned, as observed in primary visual cortex of mice. Furthermore, a novel transition between high and low dimensional activity allows for a wide and flexible tuning of dimensionality, as observed throughout cortex. The here discovered structures thus largely complement motifs. In fact, they are of a different, collective nature: they are not reflected by any local motif configuration. This result shows that many functionally relevant structures can remain hidden within the apparent randomness of highly heterogeneous cortical circuits. Our methods enable revealing these structures and investigate their effect on network dynamics.

cond-mat.dis-nn↗

arXiv2023

Field theory for optimal signal propagation in ResNets

Residual networks have significantly better trainability and thus performance than feed-forward networks at large depth. Introducing skip connections facilitates signal propagation to deeper layers. In addition, previous works found that adding a scaling parameter for the residual branch further improves generalization performance. While they empirically identified a particularly beneficial range of values for this scaling parameter, the associated performance improvement and its universality across network hyperparameters yet need to be understood. For feed-forward networks, finite-size theories have led to important insights with regard to signal propagation and hyperparameter tuning. We here derive a systematic finite-size field theory for residual networks to study signal propagation and its dependence on the scaling for the residual branch. We derive analytical expressions for the response function, a measure for the network's sensitivity to inputs, and show that for deep networks the empirically found values for the scaling parameter lie within the range of maximal sensitivity. Furthermore, we obtain an analytical expression for the optimal scaling parameter that depends only weakly on other network hyperparameters, such as the weight variance, thereby explaining its universality across hyperparameters. Overall, this work provides a theoretical framework to study ResNets at finite size.

cond-mat.dis-nn↗

arXiv2023

Learning Interacting Theories from Data

One challenge of physics is to explain how collective properties arise from microscopic interactions. Indeed, interactions form the building blocks of almost all physical theories and are described by polynomial terms in the action. The traditional approach is to derive these terms from elementary processes and then use the resulting model to make predictions for the entire system. But what if the underlying processes are unknown? Can we reverse the approach and learn the microscopic action by observing the entire system? We use invertible neural networks (INNs) to first learn the observed data distribution. By the choice of a suitable nonlinearity for the neuronal activation function, we are then able to compute the action from the weights of the trained model; a diagrammatic language expresses the change of the action from layer to layer. This process uncovers how the network hierarchically constructs interactions via nonlinear transformations of pairwise relations. We test this approach on simulated data sets of interacting theories. The network consistently reproduces a broad class of unimodal distributions; outside this class, it finds effective theories that approximate the data statistics up to the third cumulant. We explicitly show how network depth and data quantity jointly improve the agreement between the learned and the true model. This work shows how to leverage the power of machine learning to transparently extract microscopic models from data.

cond-mat.dis-nn↗

arXiv2022

Decomposing neural networks as mappings of correlation functions

Understanding the functional principles of information processing in deep neural networks continues to be a challenge, in particular for networks with trained and thus non-random weights. To address this issue, we study the mapping between probability distributions implemented by a deep feed-forward network. We characterize this mapping as an iterated transformation of distributions, where the non-linearity in each layer transfers information between different orders of correlation functions. This allows us to identify essential statistics in the data, as well as different information representations that can be used by neural networks. Applied to an XOR task and to MNIST, we show that correlations up to second order predominantly capture the information processing in the internal layers, while the input layer also extracts higher-order correlations from the data. This analysis provides a quantitative and explainable perspective on classification.

cond-mat.dis-nn↗

arXiv2022

Unified field theoretical approach to deep and recurrent neuronal networks

Understanding capabilities and limitations of different network architectures is of fundamental importance to machine learning. Bayesian inference on Gaussian processes has proven to be a viable approach for studying recurrent and deep networks in the limit of infinite layer width, $n\to\infty$. Here we present a unified and systematic derivation of the mean-field theory for both architectures that starts from first principles by employing established methods from statistical physics of disordered systems. The theory elucidates that while the mean-field equations are different with regard to their temporal structure, they yet yield identical Gaussian kernels when readouts are taken at a single time point or layer, respectively. Bayesian inference applied to classification then predicts identical performance and capabilities for the two architectures. Numerically, we find that convergence towards the mean-field theory is typically slower for recurrent networks than for deep networks and the convergence speed depends non-trivially on the parameters of the weight prior as well as the depth or number of time steps, respectively. Our method exposes that Gaussian processes are but the lowest order of a systematic expansion in $1/n$ and we compute next-to-leading-order corrections which turn out to be architecture-specific. The formalism thus paves the way to investigate the fundamental differences between recurrent and deep architectures at finite widths $n$.

cond-mat.dis-nn↗

arXiv2022

Self-consistent formulations for stochastic nonlinear neuronal dynamics

Neural dynamics is often investigated with tools from bifurcation theory. However, many neuron models are stochastic, mimicking fluctuations in the input from unknown parts of the brain or the spiking nature of signals. Noise changes the dynamics with respect to the deterministic model; in particular bifurcation theory cannot be applied. We formulate stochastic neuronal dynamics in the Martin-Siggia-Rose de Dominicis-Janssen (MSRDJ) formalism and present the fluctuation expansion of the effective action and the functional renormalization group (fRG) as two systematic ways to incorporate corrections to the mean dynamics and time-dependent statistics due to fluctuations in the presence of nonlinear neuronal gain. To formulate self-consistency equations, we derive a fundamental link between the effective action in the Onsager-Machlup(OM) formalism, which allows the study of phase transitions, and the MSRDJ effective action, which is computationally advantageous. These results in particular allow the derivation of an OM effective action for systems with non-Gaussian noise. This approach naturally leads to effective deterministic equations for the first moment of the stochastic system; they explain how nonlinearities and noise cooperate to produce memory effects. Moreover, the MSRDJ formulation yields an effective linear system that has identical power spectra and linear response. Starting from the better known loopwise approximation, we then discuss the use of the fRG as a method to obtain self-consistency beyond the mean. We present a new efficient truncation scheme for the hierarchy of flow equations for the vertex functions by adapting the Blaizot, Méndez and Wschebor approximation from the derivative expansion to the vertex expansion. The methods are presented by means of the simplest possible example of a stochastic differential equation that has generic features of neuronal dynamics.

cond-mat.stat-mech↗

arXiv2021

Gell-Mann-Low criticality in neural networks

Criticality is deeply related to optimal computational capacity. The lack of a renormalized theory of critical brain dynamics, however, so far limits insights into this form of biological information processing to mean-field results. These methods neglect a key feature of critical systems: the interaction between degrees of freedom across all length scales, which allows for complex nonlinear computation. We present a renormalized theory of a prototypical neural field theory, the stochastic Wilson-Cowan equation. We compute the flow of couplings, which parameterize interactions on increasing length scales. Despite similarities with the Kardar-Parisi-Zhang model, the theory is of a Gell-Mann-Low type, the archetypal form of a renormalizable quantum field theory. Here, nonlinear couplings vanish, flowing towards the Gaussian fixed point, but logarithmically slowly, thus remaining effective on most scales. We show this critical structure of interactions to implement a desirable trade-off between linearity, optimal for information storage, and nonlinearity, required for computation.

cond-mat.dis-nn↗

arXiv2021

Transient chaotic dimensionality expansion by recurrent networks

Neurons in the brain communicate with spikes, which are discrete events in time and value. Functional network models often employ rate units that are continuously coupled by analog signals. Is there a qualitative difference implied by these two forms of signaling? We develop a unified mean-field theory for large random networks to show that first- and second-order statistics in rate and binary networks are in fact identical if rate neurons receive the right amount of noise. Their response to presented stimuli, however, can be radically different. We quantify these differences by studying how nearby state trajectories evolve over time, asking to what extent the dynamics is chaotic. Chaos in the two models is found to be qualitatively different. In binary networks we find a network-size-dependent transition to chaos and a chaotic submanifold whose dimensionality expands stereotypically with time, while rate networks with matched statistics are nonchaotic. Dimensionality expansion in chaotic binary networks aids classification in reservoir computing and optimal performance is reached within about a single activation per neuron; a fast mechanism for computation that we demonstrate also in spiking networks. A generalization of this mechanism extends to rate networks in their respective chaotic regimes.

cond-mat.dis-nn↗

arXiv2021

Event-based update of synapses in voltage-based learning rules

Due to the point-like nature of neuronal spiking, efficient neural network simulators often employ event-based simulation schemes for synapses. Yet many types of synaptic plasticity rely on the membrane potential of the postsynaptic cell as a third factor in addition to pre- and postsynaptic spike times. Synapses therefore require continuous information to update their strength which a priori necessitates a continuous update in a time-driven manner. The latter hinders scaling of simulations to realistic cortical network sizes and relevant time scales for learning. Here, we derive two efficient algorithms for archiving postsynaptic membrane potentials, both compatible with modern simulation engines based on event-based synapse updates. We theoretically contrast the two algorithms with a time-driven synapse update scheme to analyze advantages in terms of memory and computations. We further present a reference implementation in the spiking neural network simulator NEST for two prototypical voltage-based plasticity rules: the Clopath rule and the Urbanczik-Senn rule. For both rules, the two event-based algorithms significantly outperform the time-driven scheme. Depending on the amount of data to be stored for plasticity, which heavily differs between the rules, a strong performance increase can be achieved by compressing or sampling of information on membrane potentials. Our results on computational efficiency related to archiving of information provide guidelines for the design of learning rules in order to make them practically usable in large-scale networks.

q-bio.NC↗

arXiv2020

Unfolding recurrence by Green's functions for optimized reservoir computing

Cortical networks are strongly recurrent, and neurons have intrinsic temporal dynamics. This sets them apart from deep feed-forward networks. Despite the tremendous progress in the application of feed-forward networks and their theoretical understanding, it remains unclear how the interplay of recurrence and non-linearities in recurrent cortical networks contributes to their function. The purpose of this work is to present a solvable recurrent network model that links to feed forward networks. By perturbative methods we transform the time-continuous, recurrent dynamics into an effective feed-forward structure of linear and non-linear temporal kernels. The resulting analytical expressions allow us to build optimal time-series classifiers from random reservoir networks. Firstly, this allows us to optimize not only the readout vectors, but also the input projection, demonstrating a strong potential performance gain. Secondly, the analysis exposes how the second order stimulus statistics is a crucial element that interacts with the non-linearity of the dynamics and boosts performance.

cond-mat.dis-nn↗