Daily Papers

Understanding the properties of neural networks trained via stochastic gradient descent (SGD) is at the heart of the theory of deep learning. In this work, we take a mean-field view, and consider a two-layer ReLU network trained via SGD for a univariate regularized regression problem. Our main result is that SGD is biased towards a simple solution: at convergence, the ReLU network implements a piecewise linear map of the inputs, and the number of "knot" points - i.e., points where the tangent of the ReLU network estimator changes - between two consecutive training inputs is at most three. In particular, as the number of neurons of the network grows, the SGD dynamics is captured by the solution of a gradient flow and, at convergence, the distribution of the weights approaches the unique minimizer of a related free energy, which has a Gibbs form. Our key technical contribution consists in the analysis of the estimator resulting from this minimizer: we show that its second derivative vanishes everywhere, except at some specific locations which represent the "knot" points. We also provide empirical evidence that knots at locations distinct from the data points might occur, as predicted by our theory.

A bilayer formed by stacking two distinct materials creates a moiré lattice, which can serve as a platform for novel electronic phases. In this work we study a unique example of such a system: the graphene-black phosphorus heterostructure (G/BP), which has been suggested to have an intricate band structure. Most notably, the valence band hosts a quasi-one-dimensional region in the Brillouin zone of high density of states, suggesting that various many-body electronic phases are likely to emerge. We derive an effective tight-binding model that reproduces this band structure, and explore the emergent broken-symmetry phases when interactions are introduced. Employing a mean-field analysis, we find that the favored ground-state exhibits a striped spin density wave (SDW) order, characterized by either one of two-fold degenerate wave-vectors that are tunable by gating. Further exploring the phase-diagram controlled by gate voltage and the interaction strength, we find that the SDW-ordered state undergoes a metal to insulator transition via an intermediate metallic phase which supports striped SDW correlations. Possible experimental signatures are discussed, in particular a highly anisotropic dispersion of the collective excitations which should be manifested in electric and thermal transport.

Mixture-of-Experts (MoE) layers have emerged as an important tool in scaling up modern neural networks by decoupling total trainable parameters from activated parameters in the forward pass for each token. However, sparse MoEs add complexity to training due to (i) new trainable parameters (router weights) that, like all other parameter groups, require hyperparameter (HP) tuning; (ii) new architecture scale dimensions (number of and size of experts) that must be chosen and potentially taken large. To make HP selection cheap and reliable, we propose a new parameterization for transformer models with MoE layers when scaling model width, depth, number of experts, and expert (hidden) size. Our parameterization is justified by a novel dynamical mean-field theory (DMFT) analysis. When varying different model dimensions trained at a fixed token budget, we find empirically that our parameterization enables reliable HP transfer across models from 51M to over 2B total parameters. We further take HPs identified from sweeping small models on a short token horizon to train larger models on longer horizons and report performant model behaviors.

Learning the behavior of large agent populations is an important task for numerous research areas. Although the field of multi-agent reinforcement learning (MARL) has made significant progress towards solving these systems, solutions for many agents often remain computationally infeasible and lack theoretical guarantees. Mean Field Games (MFGs) address both of these issues and can be extended to Graphon MFGs (GMFGs) to include network structures between agents. Despite their merits, the real world applicability of GMFGs is limited by the fact that graphons only capture dense graphs. Since most empirically observed networks show some degree of sparsity, such as power law graphs, the GMFG framework is insufficient for capturing these network topologies. Thus, we introduce the novel concept of Graphex MFGs (GXMFGs) which builds on the graph theoretical concept of graphexes. Graphexes are the limiting objects to sparse graph sequences that also have other desirable features such as the small world property. Learning equilibria in these games is challenging due to the rich and sparse structure of the underlying graphs. To tackle these challenges, we design a new learning algorithm tailored to the GXMFG setup. This hybrid graphex learning approach leverages that the system mainly consists of a highly connected core and a sparse periphery. After defining the system and providing a theoretical analysis, we state our learning approach and demonstrate its learning capabilities on both synthetic graphs and real-world networks. This comparison shows that our GXMFG learning algorithm successfully extends MFGs to a highly relevant class of hard, realistic learning problems that are not accurately addressed by current MARL and MFG methods.

This work builds on the theoretical frameworks presented in "Liquidity pools as mean field games: A new framework" and "Liquidity pools as mean field games with transaction costs" by the same author, where the strategic interactions among traders in a constant-product market-making protocol were modelled using mean field games (MFG), first without transaction costs and then incorporating them. Here we present the formulation of a more complete model that integrates three types of agents: traders, liquidity providers (LPs), and arbitrageurs. While we do not establish existence results for this general model, the formulation identifies the main technical difficulties and lays the groundwork for future work. The LP acts as a dominating player in the sense of 'Mean field games with a dominating player' by Bensoussan, Chau, and Yam: its strategy influences the mean field distribution of the traders, and the equilibrium is sought as a solution to the coupled system of three problems that constitute a Major-Minor game. The arbitrageurs operate by solving the optimization problem presented in "An analysis of uniswap markets" by Angeris et al., and their impact on the LP is captured through the loss-versus-rebalancing of "Automated market making and loss-versus-rebalancing" by Milionis et al. The material in this article should be read as an open research proposal rather than a collection of closed results.

Despite the widespread empirical success of ResNet, the generalization properties of deep ResNet are rarely explored beyond the lazy training regime. In this work, we investigate scaled ResNet in the limit of infinitely deep and wide neural networks, of which the gradient flow is described by a partial differential equation in the large-neural network limit, i.e., the mean-field regime. To derive the generalization bounds under this setting, our analysis necessitates a shift from the conventional time-invariant Gram matrix employed in the lazy training regime to a time-variant, distribution-dependent version. To this end, we provide a global lower bound on the minimum eigenvalue of the Gram matrix under the mean-field regime. Besides, for the traceability of the dynamic of Kullback-Leibler (KL) divergence, we establish the linear convergence of the empirical error and estimate the upper bound of the KL divergence over parameters distribution. Finally, we build the uniform convergence for generalization bound via Rademacher complexity. Our results offer new insights into the generalization ability of deep ResNet beyond the lazy training regime and contribute to advancing the understanding of the fundamental properties of deep neural networks.

Effective macroeconomic policies play a crucial role in promoting economic growth and social stability. This paper models the optimal macroeconomic policy problem based on the Stackelberg Mean Field Game (SMFG), where the government acts as the leader in policy-making, and large-scale households dynamically respond as followers. This modeling method captures the asymmetric dynamic game between the government and large-scale households, and interpretably evaluates the effects of macroeconomic policies based on microfoundations, which is difficult for existing methods to achieve. We also propose a solution for SMFGs, incorporating pre-training on real data and a model-free Stackelberg mean-field reinforcement learning (SMFRL) algorithm, which operates independently of prior environmental knowledge and transitions. Our experimental results showcase the superiority of the SMFG method over other economic policies in terms of performance, efficiency-equity tradeoff, and SMFG assumption analysis. This paper significantly contributes to the domain of AI for economics by providing a powerful tool for modeling and solving optimal macroeconomic policies.

We study a fully connected Hopfield-type associative memory network with online activity-dependent synaptic plasticity, where neural states and synaptic couplings coevolve during retrieval. Using the generating-functional formalism, we derive a dynamical mean-field theory (DMFT) in the large-system limit with extensively many stored random patterns, and show that the many-body dynamics reduces to an effective single-site stochastic process with colored Gaussian crosstalk noise and delayed feedback terms. Numerical solutions of the DMFT equations agree well with direct simulations. We find that moderate plasticity enlarges the basin of attraction and increases the maximum retrievable memory load by generating a positive delayed feedback that stabilizes retrieval against crosstalk noise. However, excessively strong plasticity causes the network to imprint the imperfect initial cue itself, leading to spurious attractors and degraded retrieval performance. Consequently, an optimal plasticity strength emerges from the trade-off between memory stabilization and premature cue imprinting. These results extend the DMFT description of associative memory to networks with coevolving neural and synaptic dynamics.

The entropic fictitious play (EFP) is a recently proposed algorithm that minimizes the sum of a convex functional and entropy in the space of measures -- such an objective naturally arises in the optimization of a two-layer neural network in the mean-field regime. In this work, we provide a concise primal-dual analysis of EFP in the setting where the learning problem exhibits a finite-sum structure. We establish quantitative global convergence guarantees for both the continuous-time and discrete-time dynamics based on properties of a proximal Gibbs measure introduced in Nitanda et al. (2022). Furthermore, our primal-dual framework entails a memory-efficient particle-based implementation of the EFP update, and also suggests a connection to gradient boosting methods. We illustrate the efficiency of our novel implementation in experiments including neural network optimization and image synthesis.

Recent frontier large language models predominantly rely on Mixture-of-Experts (MoE) architectures. Despite empirical progress, there is still no principled understanding of how hyperparameters should scale with network width N, expert width N_e, number of experts M, sparsity K, and depth L to ensure both stability and optimal performance at scale. We take a principled step toward resolving this gap by analyzing three different scaling regimes: (I) co-scaling Nasymp N_e, (II) co-scaling Nasymp Masymp K, and (III) full proportional scaling of N, N_e, M, and K. For each regime, we develop a novel Dynamical Mean Field Theory (DMFT) description of the limiting training dynamics of MoEs that provides a formal foundation for our analysis. Within this framework, we derive the unique parameterization for SGD and Adam satisfying all maximal-update (μ) desiderata. We then show that the resulting μP prescription does not reliably induce monotonic improvement with scale or robust learning-rate transfer. We trace these pathologies to scale-dependent observables in the aggregation dynamics, which motivates a refined set of desiderata that we term maximal scale stability. Guided by this principle, we derive a Maximally Scale-Stable Parameterization (MSSP) for both SGD and Adam in all three scaling regimes, and characterize the corresponding limiting dynamics - qualitatively distinct from the μP limit - through a separate DMFT analysis. Experiments verify that MSSP robustly recovers learning rate transfer and monotonic improvement with scale across regimes. Combined with existing depth-scaling theory, these results provide a complete scaling prescription for MoE architectures as a function of width, depth, expert width, and number of experts.

Neural ordinary differential equations (neural ODEs) have rapidly gained prominence as a powerful and unifying framework for conceptualizing artificial neural networks, elegantly connecting the continuous-time modeling of dynamical systems with the discrete, data-driven paradigm of modern deep learning. Beyond their practical advantages they offer fresh theoretical insights into the training and generalization properties of neural networks. The distinctive feature of this framework is its dual dynamical nature: inference dynamics, which govern the ODE evolution during forward computation, and training dynamics, which control the optimization of model parameters. This makes neural ODEs a particularly well-suited theoretical framework for studying a large variety of settings such as multi-layer neural networks (ResNets for example), autoregressive models (with next-token generation dynamics), generative models, and recurrent neural networks in theoretical neuroscience. In this work, we introduce a theoretically grounded class of models for studying neural ODEs trained via online stochastic gradient descent. We solve the training dynamics of these models via dynamical mean field theory and derive learning curves in the high-dimensional limit.

Machine learning algorithms relying on deep neural networks recently allowed a great leap forward in artificial intelligence. Despite the popularity of their applications, the efficiency of these algorithms remains largely unexplained from a theoretical point of view. The mathematical description of learning problems involves very large collections of interacting random variables, difficult to handle analytically as well as numerically. This complexity is precisely the object of study of statistical physics. Its mission, originally pointed towards natural systems, is to understand how macroscopic behaviors arise from microscopic laws. Mean-field methods are one type of approximation strategy developed in this view. We review a selection of classical mean-field methods and recent progress relevant for inference in neural networks. In particular, we remind the principles of derivations of high-temperature expansions, the replica method and message passing algorithms, highlighting their equivalences and complementarities. We also provide references for past and current directions of research on neural networks relying on mean-field methods.

In this paper, we explore the application of mean field theory, a technique from statistical physics, to deep metric learning and address the high training complexity commonly associated with conventional metric learning loss functions. By adapting mean field theory for deep metric learning, we develop an approach to design classification-based loss functions from pair-based ones, which can be considered complementary to the proxy-based approach. Applying the mean field theory to two pair-based loss functions, we derive two new loss functions, MeanFieldContrastive and MeanFieldClassWiseMultiSimilarity losses, with reduced training complexity. We extensively evaluate these derived loss functions on three image-retrieval datasets and demonstrate that our loss functions outperform baseline methods in two out of the three datasets.

Mean Field Variational Inference (MFVI) is widely understood to underestimate posterior variance. By analysing conjugate Bayesian Linear Regression (BLR), we show that this characterization is incomplete: while MFVI underestimates the variance in parameter space, it can overestimate the predictive variance compared to the exact posterior. We show that if the MFVI posterior underestimates predictive variances in some directions, it necessarily overestimates them in others. Crucially, this overestimation occurs in directions where the training data concentrates. This leads to the surprising result that, for a test point drawn from the training distribution, MFVI's expected predictive variance exceeds that of the exact posterior. We demonstrate a pathological case of this effect, where the MFVI posterior fails to reduce predictive variance compared to the prior on in distribution data. We connect these results to the Cold Posterior Effect, arguing that varying the temperature can correct this overestimation, yielding predictions closer to those of the exact posterior. We validate our theory on synthetic and real-world regression tasks.

We study in detail a one-dimensional lattice model of a continuum, conserved field (mass) that is transferred deterministically between neighbouring random sites. The model falls in a wider class of lattice models capturing the joint effect of random advection and diffusion and encompassing as specific cases, some models studied in the literature, like the Kang-Redner, Kipnis-Marchioro-Presutti, Takayasu-Taguchi, etc. The motivation for our setup comes from a straightforward interpretation as advection of particles in one-dimensional turbulence, but it is also related to a problem of synchronization of dynamical systems driven by common noise. For finite lattices, we study both the coalescence of an initially spread field (interpreted as roughening), and the statistical steady-state properties. We distinguish two main size-dependent regimes, depending on the strength of the diffusion term and on the lattice size. Using numerical simulations and mean-field approach, we study the statistics of the field. For weak diffusion, we unveil a characteristic hierarchical structure of the field. We also connect the model and the iterated function systems concept.

The pre-training cost of large language models (LLMs) is prohibitive. One cutting-edge approach to reduce the cost is zero-shot weight transfer, also known as model growth for some cases, which magically transfers the weights trained in a small model to a large model. However, there are still some theoretical mysteries behind the weight transfer. In this paper, inspired by prior applications of mean field theory to neural network dynamics, we introduce a mean field ansatz to provide a theoretical explanation for weight transfer. Specifically, we propose the row-column (RC) ansatz under the mean field point of view, which describes the measure structure of the weights in the neural network (NN) and admits a close measure dynamic. Thus, the weights of different sizes NN admit a common distribution under proper assumptions, and weight transfer methods can be viewed as sampling methods. We empirically validate the RC ansatz by exploring simple MLP examples and LLMs such as GPT-3 and Llama-3.1. We show the mean-field point of view is adequate under suitable assumptions which can provide theoretical support for zero-shot weight transfer.

Wasserstein Gradient Flows (WGF) with respect to specific functionals have been widely used in the machine learning literature. Recently, neural networks have been adopted to approximate certain intractable parts of the underlying Wasserstein gradient flow and result in efficient inference procedures. In this paper, we introduce the Neural Sinkhorn Gradient Flow (NSGF) model, which parametrizes the time-varying velocity field of the Wasserstein gradient flow w.r.t. the Sinkhorn divergence to the target distribution starting a given source distribution. We utilize the velocity field matching training scheme in NSGF, which only requires samples from the source and target distribution to compute an empirical velocity field approximation. Our theoretical analyses show that as the sample size increases to infinity, the mean-field limit of the empirical approximation converges to the true underlying velocity field. To further enhance model efficiency on high-dimensional tasks, a two-phase NSGF++ model is devised, which first follows the Sinkhorn flow to approach the image manifold quickly (le 5 NFEs) and then refines the samples along a simple straight flow. Numerical experiments with synthetic and real-world benchmark datasets support our theoretical results and demonstrate the effectiveness of the proposed methods.

In this paper, we extend mean-field Langevin dynamics to minimax optimization over probability distributions for the first time with symmetric and provably convergent updates. We propose mean-field Langevin averaged gradient (MFL-AG), a single-loop algorithm that implements gradient descent ascent in the distribution spaces with a novel weighted averaging, and establish average-iterate convergence to the mixed Nash equilibrium. We also study both time and particle discretization regimes and prove a new uniform-in-time propagation of chaos result which accounts for the dependency of the particle interactions on all previous distributions. Furthermore, we propose mean-field Langevin anchored best response (MFL-ABR), a symmetric double-loop algorithm based on best response dynamics with linear last-iterate convergence. Finally, we study applications to zero-sum Markov games and conduct simulations demonstrating long-term optimality.

In this paper, we introduce a new class of score-based generative models (SGMs) designed to handle high-cardinality data distributions by leveraging concepts from mean-field theory. We present mean-field chaos diffusion models (MF-CDMs), which address the curse of dimensionality inherent in high-cardinality data by utilizing the propagation of chaos property of interacting particles. By treating high-cardinality data as a large stochastic system of interacting particles, we develop a novel score-matching method for infinite-dimensional chaotic particle systems and propose an approximation scheme that employs a subdivision strategy for efficient training. Our theoretical and empirical results demonstrate the scalability and effectiveness of MF-CDMs for managing large high-cardinality data structures, such as 3D point clouds.

Mean-field games have been used as a theoretical tool to obtain an approximate Nash equilibrium for symmetric and anonymous N-player games. However, limiting applicability, existing theoretical results assume variations of a "population generative model", which allows arbitrary modifications of the population distribution by the learning algorithm. Moreover, learning algorithms typically work on abstract simulators with population instead of the N-player game. Instead, we show that N agents running policy mirror ascent converge to the Nash equilibrium of the regularized game within mathcal{O}(varepsilon^{-2}) samples from a single sample trajectory without a population generative model, up to a standard O(1{N}) error due to the mean field. Taking a divergent approach from the literature, instead of working with the best-response map we first show that a policy mirror ascent map can be used to construct a contractive operator having the Nash equilibrium as its fixed point. We analyze single-path TD learning for N-agent games, proving sample complexity guarantees by only using a sample path from the N-agent simulator without a population generative model. Furthermore, we demonstrate that our methodology allows for independent learning by N agents with finite sample guarantees.

Finding the mixed Nash equilibria (MNE) of a two-player zero sum continuous game is an important and challenging problem in machine learning. A canonical algorithm to finding the MNE is the noisy gradient descent ascent method which in the infinite particle limit gives rise to the {\em Mean-Field Gradient Descent Ascent} (GDA) dynamics on the space of probability measures. In this paper, we first study the convergence of a two-scale Mean-Field GDA dynamics for finding the MNE of the entropy-regularized objective. More precisely we show that for each finite temperature (or regularization parameter), the two-scale Mean-Field GDA with a suitable {\em finite} scale ratio converges exponentially to the unique MNE without assuming the convexity or concavity of the interaction potential. The key ingredient of our proof lies in the construction of new Lyapunov functions that dissipate exponentially along the Mean-Field GDA. We further study the simulated annealing of the Mean-Field GDA dynamics. We show that with a temperature schedule that decays logarithmically in time the annealed Mean-Field GDA converges to the MNE of the original unregularized objective.

We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training mean-field neural networks, maximum mean discrepancy minimization and kernel Stein discrepancy minimization. Our algorithm is based on a novel spacetime discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.

The Mean Spherical Approximation (MSA) is a commonly-used thermodynamic theory for computing the energetics of ions in the primitive model (i.e., charged hard-sphere ions in a background dielectric). For the excess chemical potential, however, the early MSA formulations (which were widely adopted) only included the terms needed to compute the mean excess chemical potential (or the mean activity coefficient). Other terms for the chemical potential μ_i of individual species i were not included because they sum to 0 in the mean chemical potential. Here, we derive these terms to give a complete MSA formulation of the chemical potential. The result is a simple additive term for μ_i that we show is a qualitative improvement over the previous MSA version. In addition, our derivation shows that the MSA's assumption of global charge neutrality is not strictly necessary, so that the MSA is also valid for systems close to neutrality.

Coordinating large populations of interacting agents is a central challenge in multi-agent reinforcement learning (MARL), where the size of the joint state-action space scales exponentially with the number of agents. Mean-field methods alleviate this burden by aggregating agent interactions, but these approaches assume homogeneous interactions. Recent graphon-based frameworks capture heterogeneity, but are computationally expensive as the number of agents grows. Therefore, we introduce GMFS, a Graphon Mean-Field Subsampling framework for scalable cooperative MARL with heterogeneous agent interactions. By subsampling κ agents according to interaction strength, we approximate the graphon-weighted mean-field and learn a policy with sample complexity poly(κ) and optimality gap O(1/sqrtκ). We verify our theory with numerical simulations in robotic coordination, showing that GMFS achieves near-optimal performance.

We propose that the grokking phenomenon, where the train loss of a neural network decreases much earlier than its test loss, can arise due to a neural network transitioning from lazy training dynamics to a rich, feature learning regime. To illustrate this mechanism, we study the simple setting of vanilla gradient descent on a polynomial regression problem with a two layer neural network which exhibits grokking without regularization in a way that cannot be explained by existing theories. We identify sufficient statistics for the test loss of such a network, and tracking these over training reveals that grokking arises in this setting when the network first attempts to fit a kernel regression solution with its initial features, followed by late-time feature learning where a generalizing solution is identified after train loss is already low. We provide an asymptotic theoretical description of the grokking dynamics in this model using dynamical mean field theory (DMFT) for high dimensional data. We find that the key determinants of grokking are the rate of feature learning -- which can be controlled precisely by parameters that scale the network output -- and the alignment of the initial features with the target function y(x). We argue this delayed generalization arises when (1) the top eigenvectors of the initial neural tangent kernel and the task labels y(x) are misaligned, but (2) the dataset size is large enough so that it is possible for the network to generalize eventually, but not so large that train loss perfectly tracks test loss at all epochs, and (3) the network begins training in the lazy regime so does not learn features immediately. We conclude with evidence that this transition from lazy (linear model) to rich training (feature learning) can control grokking in more general settings, like on MNIST, one-layer Transformers, and student-teacher networks.

The cost of hyperparameter tuning in deep learning has been rising with model sizes, prompting practitioners to find new tuning methods using a proxy of smaller networks. One such proposal uses muP parameterized networks, where the optimal hyperparameters for small width networks transfer to networks with arbitrarily large width. However, in this scheme, hyperparameters do not transfer across depths. As a remedy, we study residual networks with a residual branch scale of 1/text{depth} in combination with the muP parameterization. We provide experiments demonstrating that residual architectures including convolutional ResNets and Vision Transformers trained with this parameterization exhibit transfer of optimal hyperparameters across width and depth on CIFAR-10 and ImageNet. Furthermore, our empirical findings are supported and motivated by theory. Using recent developments in the dynamical mean field theory (DMFT) description of neural network learning dynamics, we show that this parameterization of ResNets admits a well-defined feature learning joint infinite-width and infinite-depth limit and show convergence of finite-size network dynamics towards this limit.

Deep Boltzmann machines (DBMs), one of the first ``deep'' learning methods ever studied, are multi-layered probabilistic models governed by a pairwise energy function that describes the likelihood of all variables/nodes in the network. In practice, DBMs are often constrained, i.e., via the restricted Boltzmann machine (RBM) architecture (which does not permit intra-layer connections), in order to allow for more efficient inference. In this work, we revisit the generic DBM approach, and ask the question: are there other possible restrictions to their design that would enable efficient (approximate) inference? In particular, we develop a new class of restricted model, the monotone DBM, which allows for arbitrary self-connection in each layer, but restricts the weights in a manner that guarantees the existence and global uniqueness of a mean-field fixed point. To do this, we leverage tools from the recently-proposed monotone Deep Equilibrium model and show that a particular choice of activation results in a fixed-point iteration that gives a variational mean-field solution. While this approach is still largely conceptual, it is the first architecture that allows for efficient approximate inference in fully-general weight structures for DBMs. We apply this approach to simple deep convolutional Boltzmann architectures and demonstrate that it allows for tasks such as the joint completion and classification of images, within a single deep probabilistic setting, while avoiding the pitfalls of mean-field inference in traditional RBMs.

Biological neural networks can operate in qualitatively distinct dynamical regimes, and transitions between these regimes are thought to underlie changes in computation and behavior. The seminal work of Sompolinsky, Crisanti, and Sommers (SCS) showed that random recurrent networks undergo a transition from quiescence to asynchronous chaos, establishing a paradigmatic link between random connectivity, dynamical instability, and internally generated fluctuations in neural circuits. Here, we extend this framework to two-population firing-rate networks with segregated excitatory and inhibitory neurons and target-specific inhibitory couplings that break excitation--inhibition balance. Using dynamical mean-field theory, we derive self-consistent equations for the macroscopic mean activities and autocorrelations, together with stability criteria distinguishing mean-driven and fluctuation-driven instabilities. We show that target-specific inhibition organizes the phase diagram into three qualitative classes: inhibition-dominated or strictly balanced networks display only quiescent activity and asynchronous chaos; excitation-dominated networks display persistent activity together with either synchronous chaos with non-vanishing mean activity or coherent oscillations, depending on the stability-matrix eigenvalues. Crucially, coherent oscillations do not coexist with chaotic fluctuations around the periodic mean trajectory; rather, their onset suppresses the chaotic component, reminiscent of input-induced suppression of chaos. These results generalize SCS theory to recurrent networks with explicit excitatory--inhibitory structure and identify target-specific inhibition as a key control parameter for large-scale neural dynamics.

Motivated by variational inference methods, we propose a zeroth-order algorithm for solving optimization problems in the space of Gaussian probability measures. The algorithm is based on an interacting system of Gaussian particles that stochastically explore the search space and self-organize around global minima via a consensus-based optimization (CBO) mechanism. Its construction relies on the Linearized Bures-Wasserstein (LBW) space, a novel parametrization of Gaussian measures we introduce for efficient computations. LBW is inspired by linearized optimal transport and preserves key geometric features while enabling computational tractability. We establish well-posedness and study the convergence properties of the particle dynamics via a mean-field approximation. Numerical experiments on variational inference tasks demonstrate the algorithm's robustness and superior performance with respect to deterministic gradient-based method in presence of low-dimensional non log-concave targets.

The forward pass of a Transformer can be seen as an interacting particle system on the unit sphere: time plays the role of layers, particles that of token embeddings, and the unit sphere idealizes layer normalization. In some weight settings the system can even be seen as a gradient flow for an explicit energy, and one can make sense of the infinite context length (mean-field) limit thanks to Wasserstein gradient flows. In this paper we study the effect of the perceptron block in this setting, and show that critical points are generically atomic and localized on subsets of the sphere.

Approximations based on moment-closure (MA) are commonly used to obtain estimates of the mean molecule numbers and of the variance of fluctuations in the number of molecules of chemical systems. The advantage of this approach is that it can be far less computationally expensive than exact stochastic simulations of the chemical master equation. Here we numerically study the conditions under which the MA equations yield results reflecting the true stochastic dynamics of the system. We show that for bistable and oscillatory chemical systems with deterministic initial conditions, the solution of the MA equations can be interpreted as a valid approximation to the true moments of the CME, only when the steady-state mean molecule numbers obtained from the chemical master equation fall within a certain finite range. The same validity criterion for monostable systems implies that the steady-state mean molecule numbers obtained from the chemical master equation must be above a certain threshold. For mean molecule numbers outside of this range of validity, the MA equations lead to either qualitatively wrong oscillatory dynamics or to unphysical predictions such as negative variances in the molecule numbers or multiple steady-state moments of the stationary distribution as the initial conditions are varied. Our results clarify the range of validity of the MA approach and show that pitfalls in the interpretation of the results can only be overcome through the systematic comparison of the solutions of the MA equations of a certain order with those of higher orders.

Scaling Diffusion Transformers (DiTs) to hundreds of layers introduces a structural vulnerability: networks can enter a silent, mean-dominated collapse state that homogenizes token representations and suppresses centered variation. Through mechanistic auditing, we isolate the trigger event of this collapse as Mean Mode Screaming (MMS). MMS can occur even when training appears stable, with a mean-coherent backward shock on residual writers that opens deep residual branches and drives the network into a mean-dominated state. We show this behavior is driven by an exact decomposition of these gradients into mean-coherent and centered components, compounded by the structural suppression of attention-logit gradients through the null space of the Softmax Jacobian once values homogenize. To address this, we propose Mean-Variance Split (MV-Split) Residuals, which combine a separately gained centered residual update with a leaky trunk-mean replacement. On a 400-layer single-stream DiT, MV-Split prevents the divergent collapse that crashes the un-stabilized baseline; it tracks close to the baseline's pre-crash trajectory while remaining substantially better than token-isotropic gating methods such as LayerScale across the full schedule. Finally, we present a 1000-layer DiT as a scale-validation run at boundary scales, establishing that the architecture remains stably trainable at extreme depth.

We develop a dynamical mean-field theory for random recurrent networks with low-rank structure and firing-rate-driven adaptation. When the random connectivity is strong enough to generate chaos, increasing adaptation strength drives the network through four regimes: a static coherent state, noise-sustained oscillations that progress from regular to irregular, stochastic switching between symmetric wells, and a global limit cycle. The theory identifies two instability mechanisms, chaos onset from the random connectivity and a Hopf bifurcation of the coherent mode, and shows how adaptation shapes both through the frequency-dependent single-neuron transfer function. A reduced three-dimensional model captures the bifurcation structure of the full network. Above the chaos threshold, coherent population-level oscillations coexist with heterogeneous firing rates and network-generated stochasticity at the single-neuron level. The interaction of adaptation with random and low-rank connectivity produces a rich oscillatory repertoire, including waxing-and-waning rhythmic episodes, persistent state switching, and slow Up-Down alternations, dynamics that have been observed during wakefulness, sleep, and anesthesia.

Modern deep neural networks (DNNs) represent a formidable challenge for theorists: according to the commonly accepted probabilistic framework that describes their performance, these architectures should overfit due to the huge number of parameters to train, but in practice they do not. Here we employ results from replica mean field theory to compute the generalisation gap of machine learning models with quenched features, in the teacher-student scenario and for regression problems with quadratic loss function. Notably, this framework includes the case of DNNs where the last layer is optimised given a specific realisation of the remaining weights. We show how these results -- combined with ideas from statistical learning theory -- provide a stringent asymptotic upper bound on the generalisation gap of fully trained DNN as a function of the size of the dataset P. In particular, in the limit of large P and N_{out} (where N_out is the size of the last layer) and N_out ll P, the generalisation gap approaches zero faster than 2 N_out/P, for any choice of both architecture and teacher function. Notably, this result greatly improves existing bounds from statistical learning theory. We test our predictions on a broad range of architectures, from toy fully-connected neural networks with few hidden layers to state-of-the-art deep convolutional neural networks.

MeanFlow (MF) has recently been established as a framework for one-step generative modeling. However, its ``fastforward'' nature introduces key challenges in both the training objective and the guidance mechanism. First, the original MF's training target depends not only on the underlying ground-truth fields but also on the network itself. To address this issue, we recast the objective as a loss on the instantaneous velocity v, re-parameterized by a network that predicts the average velocity u. Our reformulation yields a more standard regression problem and improves the training stability. Second, the original MF fixes the classifier-free guidance scale during training, which sacrifices flexibility. We tackle this issue by formulating guidance as explicit conditioning variables, thereby retaining flexibility at test time. The diverse conditions are processed through in-context conditioning, which reduces model size and benefits performance. Overall, our improved MeanFlow (iMF) method, trained entirely from scratch, achieves 1.72 FID with a single function evaluation (1-NFE) on ImageNet 256times256. iMF substantially outperforms prior methods of this kind and closes the gap with multi-step methods while using no distillation. We hope our work will further advance fastforward generative modeling as a stand-alone paradigm.

In this paper we study the out-of-equilibrium phase diagram of the quantum version of Derrida's Random Energy Model, which is the simplest model of mean-field spin glasses. We interpret its corresponding quantum dynamics in Fock space as a one-particle problem in very high dimension to which we apply different theoretical methods tailored for high-dimensional lattices: the Forward-Scattering Approximation, a mapping to the Rosenzweig-Porter model, and the cavity method. Our results indicate the existence of two transition lines and three distinct dynamical phases: a completely many-body localized phase at low energy, a fully ergodic phase at high energy, and a multifractal "bad metal" phase at intermediate energy. In the latter, eigenfunctions occupy a diverging volume, yet an exponentially vanishing fraction of the total Hilbert space. We discuss the limitations of our approximations and the relationship with previous studies.

This article considers the popular MCMC method of unadjusted Langevin Monte Carlo (LMC) and provides a non-asymptotic analysis of its sampling error in 2-Wasserstein distance. The proof is based on a refinement of mean-square analysis in Li et al. (2019), and this refined framework automates the analysis of a large class of sampling algorithms based on discretizations of contractive SDEs. Using this framework, we establish an O(d/epsilon) mixing time bound for LMC, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-order derivative of the potential of target measures. This bound improves the best previously known O(d/epsilon) result and is optimal (in terms of order) in both dimension d and accuracy tolerance epsilon for target measures satisfying the aforementioned assumptions. Our theoretical analysis is further validated by numerical experiments.

Minkowski functionals quantify the morphology of smooth random fields. They are widely used to probe statistical properties of cosmological fields. Analytic formulae for ensemble expectations of Minkowski functionals are well known for Gaussian and mildly non-Gaussian fields. In this paper we extend the formulae to composite fields which are sums of two fields and explicitly derive the expressions for the sum of uncorrelated mildly non-Gaussian and Gaussian fields. These formulae are applicable to observed data which is usually a sum of the true signal and one or more secondary fields that can be either noise, or some residual contaminating signal. Our formulae provide explicit quantification of the effect of the secondary field on the morphology and statistical nature of the true signal. As examples, we apply the formulae to determine how the presence of Gaussian noise can bias the morphological properties and statistical nature of Gaussian and non-Gaussian CMB temperature maps.

We present developments and applications for the diagonalization of shell-model hamiltonians in a discrete non-orthogonal basis (DNO-SM). The method, and its actual numerical implementation CARINA, based on mean-field and beyond-mean field techniques has already been applied in previous studies and is focused on basis states selection optimization. The method is benchmarked against a full set of sd shell exact diagonalizations, and is applied for the first time to the heavy deformed ^{254}No nucleus.

The k-means++ seeding algorithm (Arthur & Vassilvitskii, 2007) is widely used in practice for the k-means clustering problem where the goal is to cluster a dataset X subset R ^d into k clusters. The popularity of this algorithm is due to its simplicity and provable guarantee of being O(log k) competitive with the optimal solution in expectation. However, its running time is O(|X|kd), making it expensive for large datasets. In this work, we present a simple and effective rejection sampling based approach for speeding up k-means++. Our first method runs in time O(nnz (X) + beta k^2d) while still being O(log k ) competitive in expectation. Here, beta is a parameter which is the ratio of the variance of the dataset to the optimal k-means cost in expectation and O hides logarithmic factors in k and |X|. Our second method presents a new trade-off between computational cost and solution quality. It incurs an additional scale-invariant factor of k^{-Omega( m/beta)} Var (X) in addition to the O(log k) guarantee of k-means++ improving upon a result of (Bachem et al, 2016a) who get an additional factor of m^{-1}Var(X) while still running in time O(nnz(X) + mk^2d). We perform extensive empirical evaluations to validate our theoretical results and to show the effectiveness of our approach on real datasets.

In this note, we show that the Local Molecular Field theory of Weeks et. al. can be re-derived as an extremum problem for an approximate Helmholtz free energy. Using the resulting free energy as a classical, fluid density functional yields an implicit solvent method identical in form to the Molecular Density Functional theory of Borgis et. al., but with an explicit formula for the 'ideal' free energy term. This new expression for the ideal free energy term can be computed from all-atom molecular dynamics of a solvent with only short-range interactions. The key hypothesis required to make the theory valid is that all smooth (and hence long-range) energy functions obey Gaussian statistics. This is essentially a random phase approximation for perturbations from a short-range only, 'reference,' fluid. This single hypothesis is enough to prove that the self-consistent LMF procedure minimizes a novel density functional whose 'ideal' free energy is the molecular system under a specific, reference Hamiltonian, as opposed to the non-interacting gas of conventional density functionals. Implementation of this new functional into existing software should be straightforward and robust.

The precise equivalence between discretized Euclidean field theories and a certain class of probabilistic graphical models, namely the mathematical framework of Markov random fields, opens up the opportunity to investigate machine learning from the perspective of quantum field theory. In this contribution we will demonstrate, through the Hammersley-Clifford theorem, that the φ^{4} scalar field theory on a square lattice satisfies the local Markov property and can therefore be recast as a Markov random field. We will then derive from the φ^{4} theory machine learning algorithms and neural networks which can be viewed as generalizations of conventional neural network architectures. Finally, we will conclude by presenting applications based on the minimization of an asymmetric distance between the probability distribution of the φ^{4} machine learning algorithms and target probability distributions.

Primordial Magnetic Fields (PMFs) -- magnetic fields originating in the early Universe and permeating the cosmological scales today -- can explain the observed microGauss-level magnetisation of galaxies and their clusters. In light of current and upcoming all-sky radio surveys, PMFs have drawn attention not only as major candidates for explaining the large-scale magnetisation of the Universe, but also as potential probes of early-Universe physics. In this paper, using cosmological simulations coupled with light-cone analysis, we study for the first time the imprints of the PMF structure on the mean rotation measure (RM) originating in the intergalactic medium (IGM), langle RM_{IGM}rangle. We introduce a new method for producing full-sky RM_{IGM} distributions and analyse the autocorrelation of RM_{IGM} on small and large angular scales; we find that PMF structures indeed show distinct signatures. The large-scale uniform model (characterised by an initially unlimited coherence scale) leads to correlations up to 90 degrees, while correlations for small-scale stochastic PMF models drop by factor of 100 at 0.17, 0.13 and 0.11 degrees angular scales, corresponding to 5.24, 4.03 and 3.52 Mpc scales (at z=2 redshift) for magnetic fields with comoving 3.49, 1.81, 1.00 Mpc/h coherence scales, respectively; the correlation amplitude of the PMF model with comoving sim 19 Mpc/h coherence scale drops only by factor of 10 at 1 degree (30.6 Mpc). These results suggests that improvements in the modelling of Galactic RM will be necessary to investigate the signature of large-scale correlated PMFs. A comparison of langle RM_{IGM}rangle redshift dependence obtained from our simulations with that from the LOFAR Two-metre Sky Survey shows agreement with our previous upper limits' estimates on the PMF strength derived from RM-rms analysis.

This paper introduces a continuous-time stochastic dynamical framework for understanding how large language models (LLMs) may self-amplify latent biases or toxicity through their own chain-of-thought reasoning. The model posits an instantaneous "severity" variable x(t) in [0,1] evolving under a stochastic differential equation (SDE) with a drift term μ(x) and diffusion σ(x). Crucially, such a process can be consistently analyzed via the Fokker--Planck approach if each incremental step behaves nearly Markovian in severity space. The analysis investigates critical phenomena, showing that certain parameter regimes create phase transitions from subcritical (self-correcting) to supercritical (runaway severity). The paper derives stationary distributions, first-passage times to harmful thresholds, and scaling laws near critical points. Finally, it highlights implications for agents and extended LLM reasoning models: in principle, these equations might serve as a basis for formal verification of whether a model remains stable or propagates bias over repeated inferences.

We study mean field portfolio games under Epstein-Zin preferences, which naturally encompass the classical time-additive power utility as a special case. In a general non-Markovian framework, we establish a uniqueness result by proving a one-to-one correspondence between Nash equilibria and the solutions to a class of BSDEs. A key ingredient in our approach is a necessary stochastic maximum principle tailored to Epstein-Zin utility and a nonlinear transformation. In the deterministic setting, we further derive an explicit closed-form solution for the equilibrium investment and consumption policies.

One-step generative modeling seeks to generate high-quality data samples in a single function evaluation, significantly improving efficiency over traditional diffusion or flow-based models. In this work, we introduce Modular MeanFlow (MMF), a flexible and theoretically grounded approach for learning time-averaged velocity fields. Our method derives a family of loss functions based on a differential identity linking instantaneous and average velocities, and incorporates a gradient modulation mechanism that enables stable training without sacrificing expressiveness. We further propose a curriculum-style warmup schedule to smoothly transition from coarse supervision to fully differentiable training. The MMF formulation unifies and generalizes existing consistency-based and flow-matching methods, while avoiding expensive higher-order derivatives. Empirical results across image synthesis and trajectory modeling tasks demonstrate that MMF achieves competitive sample quality, robust convergence, and strong generalization, particularly under low-data or out-of-distribution settings.

The Transformer is widely used in natural language processing tasks. To train a Transformer however, one usually needs a carefully designed learning rate warm-up stage, which is shown to be crucial to the final performance but will slow down the optimization and bring more hyper-parameter tunings. In this paper, we first study theoretically why the learning rate warm-up stage is essential and show that the location of layer normalization matters. Specifically, we prove with mean field theory that at initialization, for the original-designed Post-LN Transformer, which places the layer normalization between the residual blocks, the expected gradients of the parameters near the output layer are large. Therefore, using a large learning rate on those gradients makes the training unstable. The warm-up stage is practically helpful for avoiding this problem. On the other hand, our theory also shows that if the layer normalization is put inside the residual blocks (recently proposed as Pre-LN Transformer), the gradients are well-behaved at initialization. This motivates us to remove the warm-up stage for the training of Pre-LN Transformers. We show in our experiments that Pre-LN Transformers without the warm-up stage can reach comparable results with baselines while requiring significantly less training time and hyper-parameter tuning on a wide range of applications.

Bayesian optimisation is a popular approach for optimising expensive black-box functions. The next location to be evaluated is selected via maximising an acquisition function that balances exploitation and exploration. Gaussian processes, the surrogate models of choice in Bayesian optimisation, are often used with a constant prior mean function equal to the arithmetic mean of the observed function values. We show that the rate of convergence can depend sensitively on the choice of mean function. We empirically investigate 8 mean functions (constant functions equal to the arithmetic mean, minimum, median and maximum of the observed function evaluations, linear, quadratic polynomials, random forests and RBF networks), using 10 synthetic test problems and two real-world problems, and using the Expected Improvement and Upper Confidence Bound acquisition functions. We find that for design dimensions ge5 using a constant mean function equal to the worst observed quality value is consistently the best choice on the synthetic problems considered. We argue that this worst-observed-quality function promotes exploitation leading to more rapid convergence. However, for the real-world tasks the more complex mean functions capable of modelling the fitness landscape may be effective, although there is no clearly optimum choice.

Distributed Mean Estimation (DME), in which n clients communicate vectors to a parameter server that estimates their average, is a fundamental building block in communication-efficient federated learning. In this paper, we improve on previous DME techniques that achieve the optimal O(1/n) Normalized Mean Squared Error (NMSE) guarantee by asymptotically improving the complexity for either encoding or decoding (or both). To achieve this, we formalize the problem in a novel way that allows us to use off-the-shelf mathematical solvers to design the quantization.

We consider dense, associative neural-networks trained with no supervision and we investigate their computational capabilities analytically, via a statistical-mechanics approach, and numerically, via Monte Carlo simulations. In particular, we obtain a phase diagram summarizing their performance as a function of the control parameters such as the quality and quantity of the training dataset and the network storage, valid in the limit of large network size and structureless datasets. Moreover, we establish a bridge between macroscopic observables standardly used in statistical mechanics and loss functions typically used in the machine learning. As technical remarks, from the analytic side, we implement large deviations and stability analysis within Guerra's interpolation to tackle the not-Gaussian distributions involved in the post-synaptic potentials while, from the computational counterpart, we insert Plefka approximation in the Monte Carlo scheme, to speed up the evaluation of the synaptic tensors, overall obtaining a novel and broad approach to investigate neural networks in general.

Likelihood-free inference methods typically make use of a distance between simulated and real data. A common example is the maximum mean discrepancy (MMD), which has previously been used for approximate Bayesian computation, minimum distance estimation, generalised Bayesian inference, and within the nonparametric learning framework. The MMD is commonly estimated at a root-m rate, where m is the number of simulated samples. This can lead to significant computational challenges since a large m is required to obtain an accurate estimate, which is crucial for parameter estimation. In this paper, we propose a novel estimator for the MMD with significantly improved sample complexity. The estimator is particularly well suited for computationally expensive smooth simulators with low- to mid-dimensional inputs. This claim is supported through both theoretical results and an extensive simulation study on benchmark simulators.

This chapter provides a pedagogical introduction and overview of spatial and temporal correlation and fluctuation effects resulting from the fundamentally stochastic kinetics underlying chemical reactions and the dynamics of populations or epidemics. After reviewing the assumptions and mean-field type approximations involved in the construction of chemical rate equations for uniform reactant densities, we first discuss spatial clustering in birth-death systems, where non-linearities are introduced through either density-limiting pair reactions, or equivalently via local imposition of finite carrying capacities. The competition of offspring production, death, and non-linear inhibition induces a population extinction threshold, which represents a non-equilibrium phase transition that separates active from absorbing states. This continuous transition is characterized by the universal scaling exponents of critical directed percolation clusters. Next we focus on the emergence of depletion zones in single-species annihilation processes and spatial population segregation with the associated reaction fronts in two-species pair annihilation. These strong (anti-)correlation effects are dynamically generated by the underlying stochastic kinetics. Finally, we address noise-induced and fluctuation-stabilized spatio-temporal patterns in basic predator-prey systems, exemplified by spreading activity fronts in the two-species Lotka-Volterra model as well as spiral structures in the May-Leonard variant of cyclically competing three-species systems akin to rock-paper-scissors games.

We develop a data-driven deep neural operator framework to approximate multiple output states for a diesel engine and generate real-time predictions with reasonable accuracy. As emission norms become more stringent, the need for fast and accurate models that enable analysis of system behavior have become an essential requirement for system development. The fast transient processes involved in the operation of a combustion engine make it difficult to develop accurate physics-based models for such systems. As an alternative to physics based models, we develop an operator-based regression model (DeepONet) to learn the relevant output states for a mean-value gas flow engine model using the engine operating conditions as input variables. We have adopted a mean-value model as a benchmark for comparison, simulated using Simulink. The developed approach necessitates using the initial conditions of the output states to predict the accurate sequence over the temporal domain. To this end, a sequence-to-sequence approach is embedded into the proposed framework. The accuracy of the model is evaluated by comparing the prediction output to ground truth generated from Simulink model. The maximum mathcal L_2 relative error observed was approximately 6.5%. The sensitivity of the DeepONet model is evaluated under simulated noise conditions and the model shows relatively low sensitivity to noise. The uncertainty in model prediction is further assessed by using a mean ensemble approach. The worst-case error at the (mu + 2sigma) boundary was found to be 12%. The proposed framework provides the ability to predict output states in real-time and enables data-driven learning of complex input-output operator mapping. As a result, this model can be applied during initial development stages, where accurate models may not be available.

In the context of C^* dynamical systems, we consider a locally compact group G acting by ^*-automorphisms on a C^* algebra U of observables, and assume a state of U that satisfies the clustering property with respect to a net of group elements of G. That is, the two-point connected correlation function vanishes in the limit on the net, when one observable is translated under the group action. Then we show that all higher order connected correlation functions (Ursell functions, or classical cumulants) and all free correlation functions (free cumulants, from free probability) vanish at the same rate in that limit. Additionally, we show that mean clustering, also called ergodicity, extends to higher order correlations. We then apply those results to equilibrium states of quantum spin lattice models. Under certain assumptions on the range of the interaction, high temperature Gibbs states are known to be exponentially clustering w.r.t. space translations. Combined with the Lieb-Robinson bound, one obtains exponential clustering for space-time translations outside the Lieb-Robinson light-cone. Therefore, by our present results, all the higher order connected and free correlation functions will vanish exponentially under space-time translations outside the Lieb-Robinson light cone, in high temperature Gibbs states. Another consequence is that their long-time averaging over a space-time ray vanishes for almost every ray velocity.

We present DYMAG, a graph neural network based on a novel form of message aggregation. Standard message-passing neural networks, which often aggregate local neighbors via mean-aggregation, can be regarded as convolving with a simple rectangular waveform which is non-zero only on 1-hop neighbors of every vertex. Here, we go beyond such local averaging. We will convolve the node features with more sophisticated waveforms generated using dynamics such as the heat equation, wave equation, and the Sprott model (an example of chaotic dynamics). Furthermore, we use snapshots of these dynamics at different time points to create waveforms at many effective scales. Theoretically, we show that these dynamic waveforms can capture salient information about the graph including connected components, connectivity, and cycle structures even with no features. Empirically, we test DYMAG on both real and synthetic benchmarks to establish that DYMAG outperforms baseline models on recovery of graph persistence, generating parameters of random graphs, as well as property prediction for proteins, molecules and materials. Our code is available at https://github.com/KrishnaswamyLab/DYMAG.

Distributed Mean Estimation (DME) is a central building block in federated learning, where clients send local gradients to a parameter server for averaging and updating the model. Due to communication constraints, clients often use lossy compression techniques to compress the gradients, resulting in estimation inaccuracies. DME is more challenging when clients have diverse network conditions, such as constrained communication budgets and packet losses. In such settings, DME techniques often incur a significant increase in the estimation error leading to degraded learning performance. In this work, we propose a robust DME technique named EDEN that naturally handles heterogeneous communication budgets and packet losses. We derive appealing theoretical guarantees for EDEN and evaluate it empirically. Our results demonstrate that EDEN consistently improves over state-of-the-art DME techniques.

Isolated quantum many-body systems which thermalize under their own dynamics are expected to act as their own thermal baths, thereby bringing their local subsystems to thermal equilibrium. Here we show that the infinite-dimensional limit of a quantum lattice model, as described by Dynamical Mean-Field theory (DMFT), provides a natural framework to understand this self-consistent thermalization process. Using the Fermi-Hubbard model as working example, we demonstrate that the emergence of a self-consistent bath thermalising the system is characterized by a sharp thermalization front, moving balistically and separating the initial condition from the long-time thermal fixed point. We characterize the full DMFT dynamics through an effective temperature for which we derive a travelling-wave equation of the Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type. This equation allows to predict the asymptotic shape of the front and its velocity, which match perfectly the full DMFT numerics. Our results provide a new angle to understand the onset of quantum thermalisation in closed isolated systems.

Algorithms based on normalizing flows are emerging as promising machine learning approaches to sampling complicated probability distributions in a way that can be made asymptotically exact. In the context of lattice field theory, proof-of-principle studies have demonstrated the effectiveness of this approach for scalar theories, gauge theories, and statistical systems. This work develops approaches that enable flow-based sampling of theories with dynamical fermions, which is necessary for the technique to be applied to lattice field theory studies of the Standard Model of particle physics and many condensed matter systems. As a practical demonstration, these methods are applied to the sampling of field configurations for a two-dimensional theory of massless staggered fermions coupled to a scalar field via a Yukawa interaction.

MeanFlow (MF) is a diffusion-motivated generative model that enables efficient few-step generation by learning long jumps directly from noise to data. In practice, it is often used as a latent MF by leveraging the pre-trained Stable Diffusion variational autoencoder (SD-VAE) for high-dimensional data modeling. However, MF training remains computationally demanding and is often unstable. During inference, the SD-VAE decoder dominates the generation cost, and MF depends on complex guidance hyperparameters for class-conditional generation. In this work, we develop an efficient training and sampling scheme for MF in the latent space of a Representation Autoencoder (RAE), where a pre-trained vision encoder (e.g., DINO) provides semantically rich latents paired with a lightweight decoder. We observe that naive MF training in the RAE latent space suffers from severe gradient explosion. To stabilize and accelerate training, we adopt Consistency Mid-Training for trajectory-aware initialization and use a two-stage scheme: distillation from a pre-trained flow matching teacher to speed convergence and reduce variance, followed by an optional bootstrapping stage with a one-point velocity estimator to further reduce deviation from the oracle mean flow. This design removes the need for guidance, simplifies training configurations, and reduces computation in both training and sampling. Empirically, our method achieves a 1-step FID of 2.03, outperforming vanilla MF's 3.43, while reducing sampling GFLOPS by 38% and total training cost by 83% on ImageNet 256. We further scale our approach to ImageNet 512, achieving a competitive 1-step FID of 3.23 with the lowest GFLOPS among all baselines. Code is available at https://github.com/sony/mf-rae.

From the dynamics of a broad class of classical mean-field glass models one may obtain a quantum model with finite zero-temperature entropy, a quantum transition at zero temperature, and a time-reparametrization (quasi-)invariance in the dynamical equations for correlations. The low eigenvalue spectrum of the resulting quantum model is directly related to the structure and exploration of metastable states in the landscape of the original classical glass model. This mapping reveals deep connections between classical glasses and the properties of SYK-like models.

We propose a principled and effective framework for one-step generative modeling. We introduce the notion of average velocity to characterize flow fields, in contrast to instantaneous velocity modeled by Flow Matching methods. A well-defined identity between average and instantaneous velocities is derived and used to guide neural network training. Our method, termed the MeanFlow model, is self-contained and requires no pre-training, distillation, or curriculum learning. MeanFlow demonstrates strong empirical performance: it achieves an FID of 3.43 with a single function evaluation (1-NFE) on ImageNet 256x256 trained from scratch, significantly outperforming previous state-of-the-art one-step diffusion/flow models. Our study substantially narrows the gap between one-step diffusion/flow models and their multi-step predecessors, and we hope it will motivate future research to revisit the foundations of these powerful models.

The ability of Variational Autoencoders to learn disentangled representations has made them appealing for practical applications. However, their mean representations, which are generally used for downstream tasks, have recently been shown to be more correlated than their sampled counterpart, on which disentanglement is usually measured. In this paper, we refine this observation through the lens of selective posterior collapse, which states that only a subset of the learned representations, the active variables, is encoding useful information while the rest (the passive variables) is discarded. We first extend the existing definition to multiple data examples and show that active variables are equally disentangled in mean and sampled representations. Based on this extension and the pre-trained models from disentanglement lib, we then isolate the passive variables and show that they are responsible for the discrepancies between mean and sampled representations. Specifically, passive variables exhibit high correlation scores with other variables in mean representations while being fully uncorrelated in sampled ones. We thus conclude that despite what their higher correlation might suggest, mean representations are still good candidates for downstream tasks applications. However, it may be beneficial to remove their passive variables, especially when used with models sensitive to correlated features.

Recent reinforcement learning (RL) methods have achieved success in various domains. However, multi-agent RL (MARL) remains a challenge in terms of decentralization, partial observability and scalability to many agents. Meanwhile, collective behavior requires resolution of the aforementioned challenges, and remains of importance to many state-of-the-art applications such as active matter physics, self-organizing systems, opinion dynamics, and biological or robotic swarms. Here, MARL via mean field control (MFC) offers a potential solution to scalability, but fails to consider decentralized and partially observable systems. In this paper, we enable decentralized behavior of agents under partial information by proposing novel models for decentralized partially observable MFC (Dec-POMFC), a broad class of problems with permutation-invariant agents allowing for reduction to tractable single-agent Markov decision processes (MDP) with single-agent RL solution. We provide rigorous theoretical results, including a dynamic programming principle, together with optimality guarantees for Dec-POMFC solutions applied to finite swarms of interest. Algorithmically, we propose Dec-POMFC-based policy gradient methods for MARL via centralized training and decentralized execution, together with policy gradient approximation guarantees. In addition, we improve upon state-of-the-art histogram-based MFC by kernel methods, which is of separate interest also for fully observable MFC. We evaluate numerically on representative collective behavior tasks such as adapted Kuramoto and Vicsek swarming models, being on par with state-of-the-art MARL. Overall, our framework takes a step towards RL-based engineering of artificial collective behavior via MFC.

Recent breakthroughs in synaptic-resolution network connectomics have revealed that brain circuits feature fine-scale structural connectivity, such as pairs of correlated synaptic couplings known as second-order motifs. Large-scale recordings of neuronal activity in networks containing nonlinear neurons reveal macroscopic heterogeneous population dynamics throughout the brain. These findings rekindle the inquiry into this intriguing question: Can microscale synaptic structures contribute to macroscopic heterogeneous dynamics and computations in ways that canonical brain circuit models cannot? To answer this question, we create random RNNs with various cell types, nonlinear non-negative neural responses, and arbitrary marginal and second-order correlated synaptic statistics. We derive mean-field low-rank equations for P-population networks in which the pre- and postsynaptic neuronal population identities determine the synaptic and motif strengths. Our framework requires 2P latent dynamic variables with P variables describing mean population activity and P variables capturing within-population variability. Theoretical and simulational results demonstrate that chain motifs induce correlations in synaptic variability, enabling microscopic fluctuations to be integrated and influence mesoscopic mean population dynamics. We apply this framework to reverse engineer network connectivity that recapitulates the heterogeneous activity across the population in the mouse primary visual cortex. By bridging the gap between synaptic organization and nonlinear heterogeneous population dynamics, our results offer a principled approach and testable predictions regarding the relationship between fine-scale connectivity, heterogeneous dynamics, and functional computations.

We establish an existence of equilibrium result for a class of non-Markovian mean-field games with unbounded control space in weak formulation. Our result is based on new existence and stability results for quadratic-growth generalized McKean-Vlasov BSDEs. Unlike earlier approaches, our approach does not require boundedness assumptions on the model parameters or time horizons and allows for running costs that are quadratic in the control variable.

We consider dense, associative neural-networks trained by a teacher (i.e., with supervision) and we investigate their computational capabilities analytically, via statistical-mechanics of spin glasses, and numerically, via Monte Carlo simulations. In particular, we obtain a phase diagram summarizing their performance as a function of the control parameters such as quality and quantity of the training dataset, network storage and noise, that is valid in the limit of large network size and structureless datasets: these networks may work in a ultra-storage regime (where they can handle a huge amount of patterns, if compared with shallow neural networks) or in a ultra-detection regime (where they can perform pattern recognition at prohibitive signal-to-noise ratios, if compared with shallow neural networks). Guided by the random theory as a reference framework, we also test numerically learning, storing and retrieval capabilities shown by these networks on structured datasets as MNist and Fashion MNist. As technical remarks, from the analytic side, we implement large deviations and stability analysis within Guerra's interpolation to tackle the not-Gaussian distributions involved in the post-synaptic potentials while, from the computational counterpart, we insert Plefka approximation in the Monte Carlo scheme, to speed up the evaluation of the synaptic tensors, overall obtaining a novel and broad approach to investigate supervised learning in neural networks, beyond the shallow limit, in general.

Clustering non-Euclidean data is difficult, and one of the most used algorithms besides hierarchical clustering is the popular algorithm Partitioning Around Medoids (PAM), also simply referred to as k-medoids clustering. In Euclidean geometry the mean-as used in k-means-is a good estimator for the cluster center, but this does not exist for arbitrary dissimilarities. PAM uses the medoid instead, the object with the smallest dissimilarity to all others in the cluster. This notion of centrality can be used with any (dis-)similarity, and thus is of high relevance to many domains and applications. A key issue with PAM is its high run time cost. We propose modifications to the PAM algorithm that achieve an O(k)-fold speedup in the second ("SWAP") phase of the algorithm, but will still find the same results as the original PAM algorithm. If we relax the choice of swaps performed (while retaining comparable quality), we can further accelerate the algorithm by eagerly performing additional swaps in each iteration. With the substantially faster SWAP, we can now explore faster initialization strategies, because (i) the classic ("BUILD") initialization now becomes the bottleneck, and (ii) our swap is fast enough to compensate for worse starting conditions. We also show how the CLARA and CLARANS algorithms benefit from the proposed modifications. While we do not study the parallelization of our approach in this work, it can easily be combined with earlier approaches to use PAM and CLARA on big data (some of which use PAM as a subroutine, hence can immediately benefit from these improvements), where the performance with high k becomes increasingly important. In experiments on real data with k=100,200, we observed a 458x respectively 1191x speedup compared to the original PAM SWAP algorithm, making PAM applicable to larger data sets, and in particular to higher k.

A model involving Gaussian processes (GPs) is introduced to simultaneously handle multi-task learning, clustering, and prediction for multiple functional data. This procedure acts as a model-based clustering method for functional data as well as a learning step for subsequent predictions for new tasks. The model is instantiated as a mixture of multi-task GPs with common mean processes. A variational EM algorithm is derived for dealing with the optimisation of the hyper-parameters along with the hyper-posteriors' estimation of latent variables and processes. We establish explicit formulas for integrating the mean processes and the latent clustering variables within a predictive distribution, accounting for uncertainty on both aspects. This distribution is defined as a mixture of cluster-specific GP predictions, which enhances the performances when dealing with group-structured data. The model handles irregular grid of observations and offers different hypotheses on the covariance structure for sharing additional information across tasks. The performances on both clustering and prediction tasks are assessed through various simulated scenarios and real datasets. The overall algorithm, called MagmaClust, is publicly available as an R package.

In natural language processing, a recently popular line of work explores how to best report the experimental results of neural networks. One exemplar publication, titled "Show Your Work: Improved Reporting of Experimental Results," advocates for reporting the expected validation effectiveness of the best-tuned model, with respect to the computational budget. In the present work, we critically examine this paper. As far as statistical generalizability is concerned, we find unspoken pitfalls and caveats with this approach. We analytically show that their estimator is biased and uses error-prone assumptions. We find that the estimator favors negative errors and yields poor bootstrapped confidence intervals. We derive an unbiased alternative and bolster our claims with empirical evidence from statistical simulation. Our codebase is at http://github.com/castorini/meanmax.

In this work, we explore two classes of density dependent relativistic mean-field models, their predictions of proton fractions at high densities and neutron star structure. We have used a metamodelling approach to these relativistic density functionals. We have generated a large ensemble of models with these classes and then applied constraints from theoretical and experimental nuclear physics and astrophysical observations. We find that both models produce similar equations of state and neutron star mass-radius sequences. But, their underlying compositions, denoted by the proton fraction in this case, are vastly different. This reinstates previous findings that information on composition gets masqueraded in beta-equilibrium. Additional observations of non-equilibrium phenomena are necessary to pin it down.

Large nonlinear recurrent neural networks with random couplings generate high-dimensional, potentially chaotic activity whose structure is of interest in neuroscience and other fields. A fundamental object encoding the collective structure of this activity is the N times N covariance matrix. Prior analytical work on the covariance matrix has been limited to low-dimensional summary statistics. Recent work proposed an ansatz in which, at large N, the covariance matrix for a typical quenched realization takes the same form as that of a linear network with the same couplings, driven by independent noise, with DMFT order parameters setting the transfer function and the noise spectrum. Here, we derive this ansatz using the two-site cavity method, providing two derivations with complementary perspectives. The first decomposes each unit's activity into a linear response to its local field and a nonlinear residual, and shows that cross-covariances between residuals at distinct sites are strongly suppressed, so the residuals act as independent noise driving a linear network. The second derives a self-consistent matrix equation for the covariance matrix. A naive Gaussian closure for the joint statistics of local fields at distinct sites misses cross terms that, in a linear network, would be generated by an external drive. The cavity method recovers these terms from non-Gaussian contributions, revealing an emergent external drive. Higher-order cross-site moments follow a Wick-like decomposition into products of pairwise covariances at leading order, reducing them to the linear-equivalent form. We verify the predictions in simulations. These results extend linear equivalence from feedforward high-dimensional nonlinear systems, where the activations are independent of the weights, to recurrent networks, where the activations are correlated with the couplings that generate them.

Integrating first-order logic constraints (FOLCs) with neural networks is a crucial but challenging problem since it involves modeling intricate correlations to satisfy the constraints. This paper proposes a novel neural layer, LogicMP, whose layers perform mean-field variational inference over an MLN. It can be plugged into any off-the-shelf neural network to encode FOLCs while retaining modularity and efficiency. By exploiting the structure and symmetries in MLNs, we theoretically demonstrate that our well-designed, efficient mean-field iterations effectively mitigate the difficulty of MLN inference, reducing the inference from sequential calculation to a series of parallel tensor operations. Empirical results in three kinds of tasks over graphs, images, and text show that LogicMP outperforms advanced competitors in both performance and efficiency.

Grid edge resources refer to distributed energy resources (DERs) located on the consumer side of the electrical grid, controlled by consumers rather than utility companies. Integrating DERs with real-time electricity pricing can better align distributed supply with system demand, improving grid efficiency and reliability. However, DER owners, known as prosumers, often lack the expertise and resources to directly participate in wholesale energy markets, limiting their ability to fully realize the economic potential of their assets. Meanwhile, as DER adoption grows, the number of prosumers participating in the energy system is expected to increase significantly, creating additional challenges in coordination and market participation. To address these challenges, we propose a mean-field game framework that enables prosumers to autonomously learn optimal decision policies based on dynamic market prices and their variable solar generation. Our framework is designed to accommodate heterogeneous agents and demonstrates the existence of a mean-field equilibrium (MFE) in a wholesale energy market with many prosumers. Additionally, we introduce an algorithm that automates prosumers' resource control, facilitating real-time decision-making for energy storage management. Numerical experiments suggest that our approach converges towards an MFE and effectively reduces peak loads and price volatility, especially during periods of external demand or supply shocks. This study highlights the potential of a fully decentralized approach to integrating DERs into wholesale markets while improving market efficiency.

We study the problem of training a flow-based generative model, parametrized by a two-layer autoencoder, to sample from a high-dimensional Gaussian mixture. We provide a sharp end-to-end analysis of the problem. First, we provide a tight closed-form characterization of the learnt velocity field, when parametrized by a shallow denoising auto-encoder trained on a finite number n of samples from the target distribution. Building on this analysis, we provide a sharp description of the corresponding generative flow, which pushes the base Gaussian density forward to an approximation of the target density. In particular, we provide closed-form formulae for the distance between the mean of the generated mixture and the mean of the target mixture, which we show decays as Theta_n(1{n}). Finally, this rate is shown to be in fact Bayes-optimal.

Large language models (LLMs) have shown remarkable progress in coding and math problem-solving, but evaluation on advanced research-level problems in hard sciences remains scarce. To fill this gap, we present CMT-Benchmark, a dataset of 50 problems covering condensed matter theory (CMT) at the level of an expert researcher. Topics span analytical and computational approaches in quantum many-body, and classical statistical mechanics. The dataset was designed and verified by a panel of expert researchers from around the world. We built the dataset through a collaborative environment that challenges the panel to write and refine problems they would want a research assistant to solve, including Hartree-Fock, exact diagonalization, quantum/variational Monte Carlo, density matrix renormalization group (DMRG), quantum/classical statistical mechanics, and model building. We evaluate LLMs by programmatically checking solutions against expert-supplied ground truth. We developed machine-grading, including symbolic handling of non-commuting operators via normal ordering. They generalize across tasks too. Our evaluations show that frontier models struggle with all of the problems in the dataset, highlighting a gap in the physical reasoning skills of current LLMs. Notably, experts identified strategies for creating increasingly difficult problems by interacting with the LLMs and exploiting common failure modes. The best model, GPT5, solves 30\% of the problems; average across 17 models (GPT, Gemini, Claude, DeepSeek, Llama) is 11.4pm2.1\%. Moreover, 18 problems are solved by none of the 17 models, and 26 by at most one. These unsolved problems span Quantum Monte Carlo, Variational Monte Carlo, and DMRG. Answers sometimes violate fundamental symmetries or have unphysical scaling dimensions. We believe this benchmark will guide development toward capable AI research assistants and tutors.

An obstacle to artificial general intelligence is set by continual learning of multiple tasks of different nature. Recently, various heuristic tricks, both from machine learning and from neuroscience angles, were proposed, but they lack a unified theory ground. Here, we focus on continual learning in single-layered and multi-layered neural networks of binary weights. A variational Bayesian learning setting is thus proposed, where the neural networks are trained in a field-space, rather than gradient-ill-defined discrete-weight space, and furthermore, weight uncertainty is naturally incorporated, and modulates synaptic resources among tasks. From a physics perspective, we translate the variational continual learning into Franz-Parisi thermodynamic potential framework, where previous task knowledge acts as a prior and a reference as well. We thus interpret the continual learning of the binary perceptron in a teacher-student setting as a Franz-Parisi potential computation. The learning performance can then be analytically studied with mean-field order parameters, whose predictions coincide with numerical experiments using stochastic gradient descent methods. Based on the variational principle and Gaussian field approximation of internal preactivations in hidden layers, we also derive the learning algorithm considering weight uncertainty, which solves the continual learning with binary weights using multi-layered neural networks, and performs better than the currently available metaplasticity algorithm. Our proposed principled frameworks also connect to elastic weight consolidation, weight-uncertainty modulated learning, and neuroscience inspired metaplasticity, providing a theory-grounded method for the real-world multi-task learning with deep networks.

Understanding the gradient variance of black-box variational inference (BBVI) is a crucial step for establishing its convergence and developing algorithmic improvements. However, existing studies have yet to show that the gradient variance of BBVI satisfies the conditions used to study the convergence of stochastic gradient descent (SGD), the workhorse of BBVI. In this work, we show that BBVI satisfies a matching bound corresponding to the ABC condition used in the SGD literature when applied to smooth and quadratically-growing log-likelihoods. Our results generalize to nonlinear covariance parameterizations widely used in the practice of BBVI. Furthermore, we show that the variance of the mean-field parameterization has provably superior dimensional dependence.

Clustering non-Euclidean data is difficult, and one of the most used algorithms besides hierarchical clustering is the popular algorithm Partitioning Around Medoids (PAM), also simply referred to as k-medoids. In Euclidean geometry the mean-as used in k-means-is a good estimator for the cluster center, but this does not hold for arbitrary dissimilarities. PAM uses the medoid instead, the object with the smallest dissimilarity to all others in the cluster. This notion of centrality can be used with any (dis-)similarity, and thus is of high relevance to many domains such as biology that require the use of Jaccard, Gower, or more complex distances. A key issue with PAM is its high run time cost. We propose modifications to the PAM algorithm to achieve an O(k)-fold speedup in the second SWAP phase of the algorithm, but will still find the same results as the original PAM algorithm. If we slightly relax the choice of swaps performed (at comparable quality), we can further accelerate the algorithm by performing up to k swaps in each iteration. With the substantially faster SWAP, we can now also explore alternative strategies for choosing the initial medoids. We also show how the CLARA and CLARANS algorithms benefit from these modifications. It can easily be combined with earlier approaches to use PAM and CLARA on big data (some of which use PAM as a subroutine, hence can immediately benefit from these improvements), where the performance with high k becomes increasingly important. In experiments on real data with k=100, we observed a 200-fold speedup compared to the original PAM SWAP algorithm, making PAM applicable to larger data sets as long as we can afford to compute a distance matrix, and in particular to higher k (at k=2, the new SWAP was only 1.5 times faster, as the speedup is expected to increase with k).

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

Normalization layers are one of the key building blocks for deep neural networks. Several theoretical studies have shown that batch normalization improves the signal propagation, by avoiding the representations from becoming collinear across the layers. However, results on mean-field theory of batch normalization also conclude that this benefit comes at the expense of exploding gradients in depth. Motivated by these two aspects of batch normalization, in this study we pose the following question: "Can a batch-normalized network keep the optimal signal propagation properties, but avoid exploding gradients?" We answer this question in the affirmative by giving a particular construction of an Multi-Layer Perceptron (MLP) with linear activations and batch-normalization that provably has bounded gradients at any depth. Based on Weingarten calculus, we develop a rigorous and non-asymptotic theory for this constructed MLP that gives a precise characterization of forward signal propagation, while proving that gradients remain bounded for linearly independent input samples, which holds in most practical settings. Inspired by our theory, we also design an activation shaping scheme that empirically achieves the same properties for certain non-linear activations.

This paper develops a new mathematical-statistical approach to analyze a class of Flajolet-Martin algorithms (FMa), and provides analytical confidence intervals for the number F0 of distinct elements in a stream, based on Chernoff bounds. The class of FMa has reached a significant popularity in bigdata stream learning, and the attention of the literature has mainly been based on algorithmic aspects, basically complexity optimality, while the statistical analysis of these class of algorithms has been often faced heuristically. The analysis provided here shows deep connections with mathematical special functions and with extreme value theory. The latter connection may help in explaining heuristic considerations, while the first opens many numerical issues, faced at the end of the present paper. Finally, the algorithms are tested on an anonymized real data stream and MonteCarlo simulations are provided to support our analytical choice in this context.

In 2000, Gillespie rehabilitated the chemical Langevin equation (CLE) by describing two conditions that must be satisfied for it yield a valid approximation of the chemical master equation (CME). In this work, we construct an original path integral description of the CME, and show how applying Gillespie's two conditions to it directly leads to a path integral equivalent to the CLE. We compare this approach to the path integral equivalent of a large system size derivation, and show that they are qualitatively different. In particular, both approaches involve converting many sums into many integrals, and the difference between the two methods is essentially the difference between using the Euler-Maclaurin formula and using Riemann sums. Our results shed light on how path integrals can be used to conceptualize coarse-graining biochemical systems, and are readily generalizable.

We characterise the dynamics of neuronal activity, in terms of field theory, using neural units placed on a 2D-lattice modelling the cortical surface. The electrical activity of neuronal units was analysed with the aim of deriving a neural field model with a simple functional form that still able to predict or reproduce empirical findings. Each neural unit was modelled using a neural mass and the accompanying field theory was derived in the continuum limit. The field theory comprised coupled (real) Klein-Gordon fields, where predictions of the model fall within the range of experimental findings. These predictions included the frequency spectrum of electric activity measured from the cortex, which was derived using an equipartition of energy over eigenfunctions of the neural fields. Moreover, the neural field model was invariant, within a set of parameters, to the dynamical system used to model each neuronal mass. Specifically, topologically equivalent dynamical systems resulted in the same neural field model when connected in a lattice; indicating that the fields derived could be read as a canonical cortical field theory. We specifically investigated non-dispersive fields that provide a structure for the coding (or representation) of afferent information. Further elaboration of the ensuing neural field theory, including the effect of dispersive forces, could be of importance in the understanding of the cortical processing of information.

Regression, the task of predicting a continuous scalar target y based on some features x is one of the most fundamental tasks in machine learning and statistics. It has been observed and theoretically analyzed that the classical approach, meansquared error minimization, can lead to suboptimal results when training neural networks. In this work, we propose a new method to improve the training of these models on regression tasks, with continuous scalar targets. Our method is based on casting this task in a different fashion, using a target encoder, and a prediction decoder, inspired by approaches in classification and clustering. We showcase the performance of our method on a wide range of real-world datasets.

The field of neuroscience and the development of artificial neural networks (ANNs) have mutually influenced each other, drawing from and contributing to many concepts initially developed in statistical mechanics. Notably, Hopfield networks and Boltzmann machines are versions of the Ising model, a model extensively studied in statistical mechanics for over a century. In the first part of this chapter, we provide an overview of the principles, models, and applications of ANNs, highlighting their connections to statistical mechanics and statistical learning theory. Artificial neural networks can be seen as high-dimensional mathematical functions, and understanding the geometric properties of their loss landscapes (i.e., the high-dimensional space on which one wishes to find extrema or saddles) can provide valuable insights into their optimization behavior, generalization abilities, and overall performance. Visualizing these functions can help us design better optimization methods and improve their generalization abilities. Thus, the second part of this chapter focuses on quantifying geometric properties and visualizing loss functions associated with deep ANNs.

Lloyd's k-means algorithm is one of the most widely used clustering methods. We prove that in high-dimensional, high-noise settings, the algorithm exhibits catastrophic failure: with high probability, essentially every partition of the data is a fixed point. Consequently, Lloyd's algorithm simply returns its initial partition - even when the underlying clusters are trivially recoverable by other methods. In contrast, we prove that Hartigan's k-means algorithm does not exhibit this pathology. Our results show the stark difference between these algorithms and offer a theoretical explanation for the empirical difficulties often observed with k-means in high dimensions.

A biological neural network in the cortex forms a neural field. Neurons in the field have their own receptive fields, and connection weights between two neurons are random but highly correlated when they are in close proximity in receptive fields. In this paper, we investigate such neural fields in a multilayer architecture to investigate the supervised learning of the fields. We empirically compare the performances of our field model with those of randomly connected deep networks. The behavior of a randomly connected network is investigated on the basis of the key idea of the neural tangent kernel regime, a recent development in the machine learning theory of over-parameterized networks; for most randomly connected neural networks, it is shown that global minima always exist in their small neighborhoods. We numerically show that this claim also holds for our neural fields. In more detail, our model has two structures: i) each neuron in a field has a continuously distributed receptive field, and ii) the initial connection weights are random but not independent, having correlations when the positions of neurons are close in each layer. We show that such a multilayer neural field is more robust than conventional models when input patterns are deformed by noise disturbances. Moreover, its generalization ability can be slightly superior to that of conventional models.

We revisit quantum phase estimation algorithms for the purpose of obtaining the energy levels of many-body Hamiltonians and pay particular attention to the statistical analysis of their outputs. We introduce the mean phase direction of the parent distribution associated with eigenstate inputs as a new post-processing tool. By connecting it with the unknown phase, we find that if used as its direct estimator, it exceeds the accuracy of the standard majority rule using one less bit of resolution, making evident that it can also be inverted to provide unbiased estimation. Moreover, we show how to directly use this quantity to accurately find the energy levels when the initialized state is an eigenstate of the simulated propagator during the whole time evolution, which allows for shallower algorithms. We then use IBM Q hardware to carry out the digital quantum simulation of three toy models: a two-level system, a two-spin Ising model and a two-site Hubbard model at half-filling. Methodologies are provided to implement Trotterization and reduce the variability of results in noisy intermediate scale quantum computers.

Modeling long-range dependencies in time series generation poses a fundamental trade-off between representational capacity and computational efficiency. Traditional temporal diffusion models suffer from local entanglement and the O(L^2) cost of attention mechanisms. We address these limitations by introducing PaCoDi (Parallel Complex Diffusion), a spectral-native architecture that decouples generative modeling in the frequency domain. PaCoDi fundamentally alters the problem topology: the Fourier Transform acts as a diagonalizing operator, converting locally coupled temporal signals into globally decorrelated spectral components. Theoretically, we prove the Quadrature Forward Diffusion and Conditional Reverse Factorization theorem, demonstrating that the complex diffusion process can be split into independent real and imaginary branches. We bridge the gap between this decoupled theory and data reality using a Mean Field Theory (MFT) approximation reinforced by an interactive correction mechanism. Furthermore, we generalize this discrete DDPM to continuous-time Frequency SDEs, rigorously deriving the Spectral Wiener Process describe the differential spectral Brownian motion limit. Crucially, PaCoDi exploits the Hermitian Symmetry of real-valued signals to compress the sequence length by half, achieving a 50% reduction in attention FLOPs without information loss. We further derive a rigorous Heteroscedastic Loss to handle the non-isotropic noise distribution on the compressed manifold. Extensive experiments show that PaCoDi outperforms existing baselines in both generation quality and inference speed, offering a theoretically grounded and computationally efficient solution for time series modeling.

We introduce ensembles within score-based sampling methods to develop gradient-free approximate sampling techniques that leverage the collective dynamics of particle ensembles to compute approximate reverse diffusion drifts. We introduce the underlying methodology, emphasizing its relationship with generative diffusion models and the previously introduced F\"ollmer sampler. We demonstrate the efficacy of ensemble strategies through various examples, ranging from low- to medium-dimensionality sampling problems, including multi-modal and highly non-Gaussian probability distributions, and provide comparisons to traditional methods like NUTS. Our findings highlight the potential of ensemble strategies for modeling complex probability distributions in situations where gradients are unavailable. Finally, we showcase its application in the context of Bayesian inversion problems within the geophysical sciences.

Neural operators, as a powerful approximation to the non-linear operators between infinite-dimensional function spaces, have proved to be promising in accelerating the solution of partial differential equations (PDE). However, it requires a large amount of simulated data, which can be costly to collect. This can be avoided by learning physics from the physics-constrained loss, which we refer to it as mean squared residual (MSR) loss constructed by the discretized PDE. We investigate the physical information in the MSR loss, which we called long-range entanglements, and identify the challenge that the neural network requires the capacity to model the long-range entanglements in the spatial domain of the PDE, whose patterns vary in different PDEs. To tackle the challenge, we propose LordNet, a tunable and efficient neural network for modeling various entanglements. Inspired by the traditional solvers, LordNet models the long-range entanglements with a series of matrix multiplications, which can be seen as the low-rank approximation to the general fully-connected layers and extracts the dominant pattern with reduced computational cost. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements from the MSR loss can be well modeled by the LordNet, yielding better accuracy and generalization ability than other neural networks. The results show that the Lordnet can be 40times faster than traditional PDE solvers. In addition, LordNet outperforms other modern neural network architectures in accuracy and efficiency with the smallest parameter size.

We revisit the theoretical properties of Hamiltonian stochastic differential equations (SDES) for Bayesian posterior sampling, and we study the two types of errors that arise from numerical SDE simulation: the discretization error and the error due to noisy gradient estimates in the context of data subsampling. Our main result is a novel analysis for the effect of mini-batches through the lens of differential operator splitting, revising previous literature results. The stochastic component of a Hamiltonian SDE is decoupled from the gradient noise, for which we make no normality assumptions. This leads to the identification of a convergence bottleneck: when considering mini-batches, the best achievable error rate is O(eta^2), with eta being the integrator step size. Our theoretical results are supported by an empirical study on a variety of regression and classification tasks for Bayesian neural networks.

Hyperparameter tuning is one of the essential steps to guarantee the convergence of machine learning models. We argue that intuition about the optimal choice of hyperparameters for stochastic gradient descent can be obtained by studying a neural network's phase diagram, in which each phase is characterised by distinctive dynamics of the singular values of weight matrices. Taking inspiration from disordered systems, we start from the observation that the loss landscape of a multilayer neural network with mean squared error can be interpreted as a disordered system in feature space, where the learnt features are mapped to soft spin degrees of freedom, the initial variance of the weight matrices is interpreted as the strength of the disorder, and temperature is given by the ratio of the learning rate and the batch size. As the model is trained, three phases can be identified, in which the dynamics of weight matrices is qualitatively different. Employing a Langevin equation for stochastic gradient descent, previously derived using Dyson Brownian motion, we demonstrate that the three dynamical regimes can be classified effectively, providing practical guidance for the choice of hyperparameters of the optimiser.

Systems of interacting objects often evolve under the influence of field effects that govern their dynamics, yet previous works have abstracted away from such effects, and assume that systems evolve in a vacuum. In this work, we focus on discovering these fields, and infer them from the observed dynamics alone, without directly observing them. We theorize the presence of latent force fields, and propose neural fields to learn them. Since the observed dynamics constitute the net effect of local object interactions and global field effects, recently popularized equivariant networks are inapplicable, as they fail to capture global information. To address this, we propose to disentangle local object interactions -- which are SE(n) equivariant and depend on relative states -- from external global field effects -- which depend on absolute states. We model interactions with equivariant graph networks, and combine them with neural fields in a novel graph network that integrates field forces. Our experiments show that we can accurately discover the underlying fields in charged particles settings, traffic scenes, and gravitational n-body problems, and effectively use them to learn the system and forecast future trajectories.

How should hidden states generated autoregressively be collapsed into a representation that reflects a language model's internal state? Despite tokens being generated under causal masking, we find that mean pooling across their hidden states yields more semantic representations than any individual token alone. We quantify this through kernel alignment to reference spaces in language, vision, and protein domains. The improvement through mean pooling is consistent with information being distributed across generated tokens rather than localized to a single position. Furthermore, representations derived from generated tokens outperform those from prompt tokens, and alignment across generation reveals interpretable dynamics in model behavior.

We study a new connection between a technical measure called mu-conductance that arises in the study of Markov chains for sampling convex bodies and the network community profile that characterizes size-resolved properties of clusters and communities in social and information networks. The idea of mu-conductance is similar to the traditional graph conductance, but disregards sets with small volume. We derive a sequence of optimization problems including a low-rank semi-definite program from which we can derive a lower bound on the optimal mu-conductance value. These ideas give the first theoretically sound bound on the behavior of the network community profile for a wide range of cluster sizes. The algorithm scales up to graphs with hundreds of thousands of nodes and we demonstrate how our framework validates the predicted structures of real-world graphs.

Personalized Federated Learning (PFL) enables a collection of agents to collaboratively learn individual models without sharing raw data. We propose a new PFL approach in which each agent optimizes a weighted combination of all agents' empirical risks, with the weights learned from data rather than specified a priori. The novelty of our method lies in formulating the estimation of these collaborative weights as a kernel mean embedding estimation problem with multiple data sources, leveraging tools from multi-task averaging to capture statistical relationships between agents. This perspective yields a fully adaptive procedure that requires no prior knowledge of data heterogeneity and can automatically transition between global and local learning regimes. By recasting the objective as a high-dimensional mean estimation problem, we derive finite-sample guarantees on local excess risks for a broad class of distributions, explicitly quantifying the statistical gains of collaboration. To address communication constraints inherent to federated settings, we also propose a practical implementation based on random Fourier features, which allows one to trade communication cost for statistical efficiency. Numerical experiments validate our theoretical results.

This paper develops a comprehensive theoretical framework that imports concepts from stochastic thermodynamics to model price impact and characterize the feasibility of round-trip arbitrage in financial markets. A trading cycle is treated as a non-equilibrium thermodynamic process, where price impact represents dissipative work and market noise plays the role of thermal fluctuations. The paper proves a Financial Second Law: under general convex impact functionals, any round-trip trading strategy yields non-positive expected profit. This structural constraint is complemented by a fluctuation theorem that bounds the probability of profitable cycles in terms of dissipated work and market volatility. The framework introduces a statistical ensemble of trading strategies governed by a Gibbs measure, leading to a free energy decomposition that connects expected cost, strategy entropy, and a market temperature parameter. The framework provides rigorous, testable inequalities linking microstructural impact to macroscopic no-arbitrage conditions, offering a novel physics-inspired perspective on market efficiency. The paper derives explicit analytical results for prototypical trading strategies and discusses empirical validation protocols.

The proper design and architecture of testing of machine learning models, especially in their application to quantitative finance problems, is crucial. The most important in this process is selecting an adequate loss function used for training, validation, estimation purposes, and tuning of hyperparameters. Therefore, in this research, through empirical experiments on equity and cryptocurrency assets, we introduce the Mean Absolute Directional Loss (MADL) function which is more adequate for optimizing forecast-generating models used in algorithmic investment strategies. The MADL function results are compared for Transformer and LSTM models and we show that almost in every case Transformer results are significantly better than those obtained with LSTM.

Machine Learning Force Fields (MLFFs) are a promising alternative to expensive ab initio quantum mechanical molecular simulations. Given the diversity of chemical spaces that are of interest and the cost of generating new data, it is important to understand how MLFFs generalize beyond their training distributions. In order to characterize and better understand distribution shifts in MLFFs, we conduct diagnostic experiments on chemical datasets, revealing common shifts that pose significant challenges, even for large foundation models trained on extensive data. Based on these observations, we hypothesize that current supervised training methods inadequately regularize MLFFs, resulting in overfitting and learning poor representations of out-of-distribution systems. We then propose two new methods as initial steps for mitigating distribution shifts for MLFFs. Our methods focus on test-time refinement strategies that incur minimal computational cost and do not use expensive ab initio reference labels. The first strategy, based on spectral graph theory, modifies the edges of test graphs to align with graph structures seen during training. Our second strategy improves representations for out-of-distribution systems at test-time by taking gradient steps using an auxiliary objective, such as a cheap physical prior. Our test-time refinement strategies significantly reduce errors on out-of-distribution systems, suggesting that MLFFs are capable of and can move towards modeling diverse chemical spaces, but are not being effectively trained to do so. Our experiments establish clear benchmarks for evaluating the generalization capabilities of the next generation of MLFFs. Our code is available at https://tkreiman.github.io/projects/mlff_distribution_shifts/.

To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Frechet mean. In this work, we equip a set of graphs with the pseudometric defined by the norm between the eigenvalues of their respective adjacency matrix. Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems for graph-valued data. We describe an algorithm to compute an approximation to the sample Frechet mean of a set of undirected unweighted graphs with a fixed size using this pseudometric.

Large Language Models achieve next-token prediction by transporting a vectorized piece of text (prompt) across an accompanying embedding space under the action of successive transformer layers. The resulting high-dimensional trajectories realize different contextualization, or 'thinking', steps, and fully determine the output probability distribution. We aim to characterize the statistical properties of ensembles of these 'lines of thought.' We observe that independent trajectories cluster along a low-dimensional, non-Euclidean manifold, and that their path can be well approximated by a stochastic equation with few parameters extracted from data. We find it remarkable that the vast complexity of such large models can be reduced to a much simpler form, and we reflect on implications.

This paper introduces a novel approach to financial risk analysis that does not rely on traditional price and market data, instead using market news to model assets as distributions over a metric space of risk factors. By representing asset returns as integrals over the scalar field of these risk factors, we derive the covariance structure between asset returns. Utilizing encoder-only language models to embed this news data, we explore the relationships between asset return distributions through the concept of Energy Distance, establishing connections between distributional differences and excess returns co-movements. This data-agnostic approach provides new insights into portfolio diversification, risk management, and the construction of hedging strategies. Our findings have significant implications for both theoretical finance and practical risk management, offering a more robust framework for modelling complex financial systems without depending on conventional market data.

Recent years have witnessed significant progress in developing efficient training and fast sampling approaches for diffusion models. A recent remarkable advancement is the use of stochastic differential equations (SDEs) to describe data perturbation and generative modeling in a unified mathematical framework. In this paper, we reveal several intriguing geometric structures of diffusion models and contribute a simple yet powerful interpretation to their sampling dynamics. Through carefully inspecting a popular variance-exploding SDE and its marginal-preserving ordinary differential equation (ODE) for sampling, we discover that the data distribution and the noise distribution are smoothly connected with an explicit, quasi-linear sampling trajectory, and another implicit denoising trajectory, which even converges faster in terms of visual quality. We also establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm, with which we can characterize the asymptotic behavior of diffusion models and identify the score deviation. These new geometric observations enable us to improve previous sampling algorithms, re-examine latent interpolation, as well as re-explain the working principles of distillation-based fast sampling techniques.

We propose an effective field-theoretic framework for analyzing Transformer attention through a thermodynamic lens. By constructing a Lagrangian on the information manifold equipped with the Fisher metric, we show that, within the Shannon--Boltzmann entropy framework, the Softmax function arises as a stationary solution minimizing a Helmholtz free energy functional. This establishes a formal correspondence between scaled dot-product attention and canonical ensemble statistics. Extending this mapping to macroscopic observables, we define an effective specific heat associated with fluctuations of the attention energy landscape. In controlled experiments on the modular addition task (p = 19--113), we observe a robust peak in this fluctuation measure that consistently precedes the onset of generalization. While no asymptotic power-law divergence is detected in this finite-depth regime, the reproducible enhancement of energy variance suggests a critical-like crossover accompanying representational reorganization. Our framework provides a unified statistical-mechanical perspective on attention scaling, training dynamics, and positional encoding, interpreting the phenomena as emergent properties of an effective thermodynamic system rather than isolated heuristics. Although the present results indicate finite-size crossover behavior rather than a strict phase transition, they motivate further investigation into scaling limits of deep architectures through fluctuation-based observables.

We develop a general mathematical framework to analyze scaling regimes and derive explicit analytic solutions for gradient flow (GF) in large learning problems. Our key innovation is a formal power series expansion of the loss evolution, with coefficients encoded by diagrams akin to Feynman diagrams. We show that this expansion has a well-defined large-size limit that can be used to reveal different learning phases and, in some cases, to obtain explicit solutions of the nonlinear GF. We focus on learning Canonical Polyadic (CP) decompositions of high-order tensors, and show that this model has several distinct extreme lazy and rich GF regimes such as free evolution, NTK and under- and over-parameterized mean-field. We show that these regimes depend on the parameter scaling, tensor order, and symmetry of the model in a specific and subtle way. Moreover, we propose a general approach to summing the formal loss expansion by reducing it to a PDE; in a wide range of scenarios, it turns out to be first-order and solvable by the method of characteristics. We observe a very good agreement of our theoretical predictions with experimental results.

Random network models play a prominent role in modeling, analyzing and understanding complex phenomena on real-life networks. However, a key property of networks is often neglected: many real-world networks exhibit spatial structure, the tendency of a node to select neighbors with a probability depending on physical distance. Here, we introduce a class of random spatial networks (RSNs) which generalizes many existing random network models but adds spatial structure. In these networks, nodes are placed randomly in space and joined in edges with a probability depending on their distance and their individual expected degrees, in a manner that crucially remains analytically tractable. We use this network class to propose a new generalization of small-world networks, where the average shortest path lengths in the graph are small, as in classical Watts-Strogatz small-world networks, but with close spatial proximity of nodes that are neighbors in the network playing the role of large clustering. Small-world effects are demonstrated on these spatial small-world networks without clustering. We are able to derive partial integro-differential equations governing susceptible-infectious-recovered disease spreading through an RSN, and we demonstrate the existence of traveling wave solutions. If the distance kernel governing edge placement decays slower than exponential, the population-scale dynamics are dominated by long-range hops followed by local spread of traveling waves. This provides a theoretical modeling framework for recent observations of how epidemics like Ebola evolve in modern connected societies, with long-range connections seeding new focal points from which the epidemic locally spreads in a wavelike manner.

Electron-electron (e-e) and electron-phonon (e-ph) interactions are challenging to describe in correlated materials, where their joint effects govern unconventional transport, phase transitions, and superconductivity. Here we combine first-principles e-ph calculations with dynamical mean field theory (DMFT) as a step toward a unified description of e-e and e-ph interactions in correlated materials. We compute the e-ph self-energy using the DMFT electron Green's function, and combine it with the e-e self-energy from DMFT to obtain a Green's function including both interactions. This approach captures the renormalization of quasiparticle dispersion and spectral weight on equal footing. Using our method, we study the e-ph and e-e contributions to the resistivity and spectral functions in the correlated metal Sr_2RuO_4. In this material, our results show that e-e interactions dominate transport and spectral broadening in the temperature range we study (50-310~K), while e-ph interactions are relatively weak and account for only sim10\% of the experimental resistivity. We also compute effective scattering rates, and find that the e-e interactions result in scattering several times greater than the Planckian value k_BT, whereas e-ph interactions are associated with scattering rates lower than k_BT. Our work demonstrates a first-principles approach to combine electron dynamical correlations from DMFT with e-ph interactions in a consistent way, advancing quantitative studies of correlated materials.

Deep Learning using the eponymous deep neural networks (DNNs) has become an attractive approach towards various data-based problems of theoretical physics in the past decade. There has been a clear trend to deeper architectures containing increasingly more powerful and involved layers. Contrarily, Taylor coefficients of DNNs still appear mainly in the light of interpretability studies, where they are computed at most to first order. However, especially in theoretical physics numerous problems benefit from accessing higher orders, as well. This gap motivates a general formulation of neural network (NN) Taylor expansions. Restricting our analysis to multilayer perceptrons (MLPs) and introducing quantities we refer to as propagators and vertices, both depending on the MLP's weights and biases, we establish a graph-theoretical approach. Similarly to Feynman rules in quantum field theories, we can systematically assign diagrams containing propagators and vertices to the corresponding partial derivative. Examining this approach for S-wave scattering lengths of shallow potentials, we observe NNs to adapt their derivatives mainly to the leading order of the target function's Taylor expansion. To circumvent this problem, we propose an iterative NN perturbation theory. During each iteration we eliminate the leading order, such that the next-to-leading order can be faithfully learned during the subsequent iteration. After performing two iterations, we find that the first- and second-order Born terms are correctly adapted during the respective iterations. Finally, we combine both results to find a proxy that acts as a machine-learned second-order Born approximation.

Transformers play a central role in the inner workings of large language models. We develop a mathematical framework for analyzing Transformers based on their interpretation as interacting particle systems, which reveals that clusters emerge in long time. Our study explores the underlying theory and offers new perspectives for mathematicians as well as computer scientists.

We carry out an information-theoretical analysis of a two-layer neural network trained from input-output pairs generated by a teacher network with matching architecture, in overparametrized regimes. Our results come in the form of bounds relating i) the mutual information between training data and network weights, or ii) the Bayes-optimal generalization error, to the same quantities but for a simpler (generalized) linear model for which explicit expressions are rigorously known. Our bounds, which are expressed in terms of the number of training samples, input dimension and number of hidden units, thus yield fundamental performance limits for any neural network (and actually any learning procedure) trained from limited data generated according to our two-layer teacher neural network model. The proof relies on rigorous tools from spin glasses and is guided by ``Gaussian equivalence principles'' lying at the core of numerous recent analyses of neural networks. With respect to the existing literature, which is either non-rigorous or restricted to the case of the learning of the readout weights only, our results are information-theoretic (i.e. are not specific to any learning algorithm) and, importantly, cover a setting where all the network parameters are trained.

In the artificial neuron, I replace the dot product with the weighted Lehmer mean, which may emulate different cases of a generalized mean. The single neuron instance is replaced by a multiplet of neurons which have the same averaging weights. A group of outputs feed forward, in lieu of the single scalar. The generalization parameter is typically set to a different value for each neuron in the multiplet. I further extend the concept to a multiplet taken from the Gini mean. Derivatives with respect to the weight parameters and with respect to the two generalization parameters are given. Some properties of the network are investigated, showing the capacity to emulate the classical exclusive-or problem organically in two layers and perform some multiplication and division. The network can instantiate truncated power series and variants, which can be used to approximate different functions, provided that parameters are constrained. Moreover, a mean case slope score is derived that can facilitate a learning-rate novelty based on homogeneity of the selected elements. The multiplet neuron equation provides a way to segment regularization timeframes and approaches.

Few-step generation has been a long-standing goal, with recent one-step generation methods exemplified by MeanFlow achieving remarkable results. Existing research on MeanFlow primarily focuses on class-to-image generation. However, an intuitive yet unexplored direction is to extend the condition from fixed class labels to flexible text inputs, enabling richer content creation. Compared to the limited class labels, text conditions pose greater challenges to the model's understanding capability, necessitating the effective integration of powerful text encoders into the MeanFlow framework. Surprisingly, although incorporating text conditions appears straightforward, we find that integrating powerful LLM-based text encoders using conventional training strategies results in unsatisfactory performance. To uncover the underlying cause, we conduct detailed analyses and reveal that, due to the extremely limited number of refinement steps in the MeanFlow generation, such as only one step, the text feature representations are required to possess sufficiently high discriminability. This also explains why discrete and easily distinguishable class features perform well within the MeanFlow framework. Guided by these insights, we leverage a powerful LLM-based text encoder validated to possess the required semantic properties and adapt the MeanFlow generation process to this framework, resulting in efficient text-conditioned synthesis for the first time. Furthermore, we validate our approach on the widely used diffusion model, demonstrating significant generation performance improvements. We hope this work provides a general and practical reference for future research on text-conditioned MeanFlow generation. The code is available at https://github.com/AMAP-ML/EMF.

Many applications involve estimating the mean of multiple binomial outcomes as a common problem -- assessing intergenerational mobility of census tracts, estimating prevalence of infectious diseases across countries, and measuring click-through rates for different demographic groups. The most standard approach is to report the plain average of each outcome. Despite simplicity, the estimates are noisy when the sample sizes or mean parameters are small. In contrast, the Empirical Bayes (EB) methods are able to boost the average accuracy by borrowing information across tasks. Nevertheless, the EB methods require a Bayesian model where the parameters are sampled from a prior distribution which, unlike the commonly-studied Gaussian case, is unidentified due to discreteness of binomial measurements. Even if the prior distribution is known, the computation is difficult when the sample sizes are heterogeneous as there is no simple joint conjugate prior for the sample size and mean parameter. In this paper, we consider the compound decision framework which treats the sample size and mean parameters as fixed quantities. We develop an approximate Stein's Unbiased Risk Estimator (SURE) for the average mean squared error given any class of estimators. For a class of machine learning-assisted linear shrinkage estimators, we establish asymptotic optimality, regret bounds, and valid inference. Unlike existing work, we work with the binomials directly without resorting to Gaussian approximations. This allows us to work with small sample sizes and/or mean parameters in both one-sample and two-sample settings. We demonstrate our approach using three datasets on firm discrimination, education outcomes, and innovation rates.

We propose a direct estimation method for R\'{e}nyi and f-divergence measures based on a new graph theoretical interpretation. Suppose that we are given two sample sets X and Y, respectively with N and M samples, where eta:=M/N is a constant value. Considering the k-nearest neighbor (k-NN) graph of Y in the joint data set (X,Y), we show that the average powered ratio of the number of X points to the number of Y points among all k-NN points is proportional to R\'{e}nyi divergence of X and Y densities. A similar method can also be used to estimate f-divergence measures. We derive bias and variance rates, and show that for the class of gamma-H\"{o}lder smooth functions, the estimator achieves the MSE rate of O(N^{-2gamma/(gamma+d)}). Furthermore, by using a weighted ensemble estimation technique, for density functions with continuous and bounded derivatives of up to the order d, and some extra conditions at the support set boundary, we derive an ensemble estimator that achieves the parametric MSE rate of O(1/N). Our estimators are more computationally tractable than other competing estimators, which makes them appealing in many practical applications.

Recent work in activation steering has demonstrated the potential to better control the outputs of Large Language Models (LLMs), but it involves finding steering vectors. This is difficult because engineers do not typically know how features are represented in these models. We seek to address this issue by applying the idea of mean-centring to steering vectors. We find that taking the average of activations associated with a target dataset, and then subtracting the mean of all training activations, results in effective steering vectors. We test this method on a variety of models on natural language tasks by steering away from generating toxic text, and steering the completion of a story towards a target genre. We also apply mean-centring to extract function vectors, more effectively triggering the execution of a range of natural language tasks by a significant margin (compared to previous baselines). This suggests that mean-centring can be used to easily improve the effectiveness of activation steering in a wide range of contexts.

The initial state of neural networks plays a central role in conditioning the subsequent training dynamics. In the context of classification problems, we provide a theoretical analysis demonstrating that the structure of a neural network can condition the model to assign all predictions to the same class, even before the beginning of training, and in the absence of explicit biases. We show that the presence of this phenomenon, which we call "Initial Guessing Bias" (IGB), depends on architectural choices such as activation functions, max-pooling layers, and network depth. Our analysis of IGB has practical consequences, in that it guides architecture selection and initialization. We also highlight theoretical consequences, such as the breakdown of node-permutation symmetry, the violation of self-averaging, the validity of some mean-field approximations, and the non-trivial differences arising with depth.

Supervised convolutional neural networks (CNNs) are widely used to solve imaging inverse problems, achieving state-of-the-art performance in numerous applications. However, despite their empirical success, these methods are poorly understood from a theoretical perspective and often treated as black boxes. To bridge this gap, we analyze trained neural networks through the lens of the Minimum Mean Square Error (MMSE) estimator, incorporating functional constraints that capture two fundamental inductive biases of CNNs: translation equivariance and locality via finite receptive fields. Under the empirical training distribution, we derive an analytic, interpretable, and tractable formula for this constrained variant, termed Local-Equivariant MMSE (LE-MMSE). Through extensive numerical experiments across various inverse problems (denoising, inpainting, deconvolution), datasets (FFHQ, CIFAR-10, FashionMNIST), and architectures (U-Net, ResNet, PatchMLP), we demonstrate that our theory matches the neural networks outputs (PSNR gtrsim25dB). Furthermore, we provide insights into the differences between physics-aware and physics-agnostic estimators, the impact of high-density regions in the training (patch) distribution, and the influence of other factors (dataset size, patch size, etc).

With the increasing volume of high-frequency data in the information age, both challenges and opportunities arise in the prediction of stock volatility. On one hand, the outcome of prediction using tradition method combining stock technical and macroeconomic indicators still leaves room for improvement; on the other hand, macroeconomic indicators and peoples' search record on those search engines affecting their interested topics will intuitively have an impact on the stock volatility. For the convenience of assessment of the influence of these indicators, macroeconomic indicators and stock technical indicators are then grouped into objective factors, while Baidu search indices implying people's interested topics are defined as subjective factors. To align different frequency data, we introduce GARCH-MIDAS model. After mixing all the above data, we then feed them into Transformer model as part of the training data. Our experiments show that this model outperforms the baselines in terms of mean square error. The adaption of both types of data under Transformer model significantly reduces the mean square error from 1.00 to 0.86.

This paper presents a novel end-to-end framework for closed-form computation and visualization of critical point uncertainty in 2D uncertain scalar fields. Critical points are fundamental topological descriptors used in the visualization and analysis of scalar fields. The uncertainty inherent in data (e.g., observational and experimental data, approximations in simulations, and compression), however, creates uncertainty regarding critical point positions. Uncertainty in critical point positions, therefore, cannot be ignored, given their impact on downstream data analysis tasks. In this work, we study uncertainty in critical points as a function of uncertainty in data modeled with probability distributions. Although Monte Carlo (MC) sampling techniques have been used in prior studies to quantify critical point uncertainty, they are often expensive and are infrequently used in production-quality visualization software. We, therefore, propose a new end-to-end framework to address these challenges that comprises a threefold contribution. First, we derive the critical point uncertainty in closed form, which is more accurate and efficient than the conventional MC sampling methods. Specifically, we provide the closed-form and semianalytical (a mix of closed-form and MC methods) solutions for parametric (e.g., uniform, Epanechnikov) and nonparametric models (e.g., histograms) with finite support. Second, we accelerate critical point probability computations using a parallel implementation with the VTK-m library, which is platform portable. Finally, we demonstrate the integration of our implementation with the ParaView software system to demonstrate near-real-time results for real datasets.