Revises mean-field theory of Santa Fe model using kinetic theory.
problem Deriving a solid mathematical foundation for the Santa Fe model.
method Systematic derivation of BBGKY hierarchy from exact master equation.
result Explicit and closed-form solutions for mean-field equations.
Federated learning linked to mean-field games for large-scale learning.
problem Large-scale distributed and privacy-preserving learning algorithms.
method Established a connection between federated learning and mean-field games, presenting federated learning as a differential game.
result Properties of the equilibrium of the federated learning game were discussed.
New algorithm recovers sparse measures in polynomial time.
problem Recovering sparse measures from Fourier moments.
method Polynomial-time recovery method inspired by mean-field theory.
result Improves upon convex relaxation methods in specific parameter regime.
Develops a dynamic mean field theory for reinforcement learning.
problem Finite state and action Bayesian reinforcement learning in large state spaces.
method Analogies with statistical physics, interpreting probabilities as couplings and values as spins, solving mean field equations.
result State-action values are statistically independent in the asymptotic state space limit, with exact or approximate equations for computation.
Mean Field Games applied to finance and economics.
problem Modeling large populations in financial and economic systems.
method Mean Field Game theory applied to specific financial and economic scenarios.
result New applications in bitcoin mining, energy markets, and macro-economics.
The paper models asset pricing in a partially observed market using mean field game theory and exponential quadratic Gaussian framework.
problem Asset pricing in a market with partial observation and heterogeneous agents.
method Mean field game theory, exponential quadratic Gaussian framework, Kalman-Bucy filtering theory.
result Characterization of equilibrium risk premium through mean field BSDE and construction of unobservable risk premium process.
Quantum field theory uses Lorentzian bordisms to describe time evolution.
problem Describing the time evolution of quantum field theories.
method Defines a functorial field theory on Lorentzian bordism pseudo-category.
result Lorentzian bordisms naturally arise in algebraic quantum field theory.
Develops asset pricing models with mean field game theory for heterogeneous agents.
problem Tackles equilibrium asset pricing in incomplete markets with heterogeneous agents.
method Uses mean field game theory and mean field backward stochastic differential equations (BSDEs).
result Derives equilibrium risk premium and shows market clearing in the large population limit.
New algorithm reduces MFGs with common noise complexity.
problem Prohibitive computational cost in solving MFGs with common noise.
method Signatured deep fictitious play based on rough path theory.
result Significantly reduced computational complexity and improved efficiency.
Analysis of SGD for Gaussian mixture classification using dynamical mean-field theory.
problem Learning dynamics of SGD for a neural network classifying Gaussian mixture.
method Applying dynamical mean-field theory to track SGD dynamics in high dimensions.
result Reveals how SGD navigates the non-convex loss landscape.
Study on kernel methods in large-scale machine learning problems.
problem Large-scale machine learning with many interacting variables.
method Mean field limit analysis of kernels and their Hilbert spaces.
result Mean field convergence of empirical and infinite-sample solutions.
Proves existence of solution to Lichnerowicz equation on non-CMC manifolds.
problem Existence of positive solution to Lichnerowicz equation on non-CMC closed manifolds with supercritical terms.
method Employed a fixed-point argument involving sub- and supersolutions, with conditions on coefficients to prevent classical solutions.
result Proves existence of a positive and essentially bounded solution.
Study explores optimal strategies in games with multiple players and mean-field interactions.
problem Optimal strategies in games with multiple players and mean-field interactions.
method Exploration of three different notions of optimality, including mean-field control solution, mean-field coarse correlated equilibria, and mean-field Nash equilibria.
result Approximation of cooperative and competitive equilibria in large N-player games by mean-field control and mean-field equilibria. Recently, variational approximations such as the mean field approximation have received much interest. We extend the standard mean field method by using an approximating distribution that factorises into cluster potentials. This includes undirected graphs, directed acyclic graphs and junction trees. We derive generaliz…
Non-linear image reconstruction and signal analysis deal with complex inverse problems. To tackle such problems in a systematic way, I present information field theory (IFT) as a means of Bayesian, data based inference on spatially distributed signal fields. IFT is a statistical field theory, which permits the construc…
Learning in restricted Boltzmann machine is typically hard due to the computation of gradients of log-likelihood function. To describe the network state statistics of the restricted Boltzmann machine, we develop an advanced mean field theory based on the Bethe approximation. Our theory provides an efficient message pas…
The paper analyzes the dynamics of a simple neural network using a mean-field approach.
problem Understanding the training dynamics of neural networks, especially in classification tasks.
method Developed an analytic theory using a mean-field limit for a simple neural network.
result Explicitly solved the dynamics of a linearly separable dataset with a linear hinge loss.
The study analyzes a three-layer neural network's training dynamics using a functional-space mean-field theory.
problem Understanding the training dynamics of partially-trained three-layer neural networks.
method Generalized mean-field theory to functional spaces, proving convergence and feature learning.
result The training loss of the model decays to zero at a linear rate in the L2 regression setting. Transformers approximate mean-field dynamics of indistinguishable particles.
problem Approximating the dynamics of indistinguishable particles in complex systems.
method Using transformers to model the mean-field dynamics of interacting particle systems.
result Theoretical bounds on the distance between true and transformer-obtained mean-field dynamics.
The paper develops a new algorithm for RBMs using dynamical mean-field theory.
problem Learning in Restricted Boltzmann Machines (RBMs) with complex dependencies.
method Dynamical mean-field theory applied to RBMs with rectangular coupling matrices drawn from a bi-rotation invariant ensemble.
result The algorithm converges globally under a stability criterion, with rates matching numerical simulations.
Improved sampling from mean-field stationary distributions.
problem Sampling from the stationary distribution of mean-field SDEs.
method Decoupling the problem into two aspects: approximation of mean-field SDE and sampling from finite-particle distribution.
result Improved guarantees in various settings, including optimizing neural networks.
Unified theory for deep and recurrent networks using Gaussian processes.
problem Understanding capabilities and limitations of different network architectures.
method Unified derivation of mean-field theory from statistical physics of disordered systems.
result Gaussian processes yield identical Gaussian kernels for both architectures at a single time point or layer.
Study of a game with multiple players and common shocks using probabilistic methods.
problem Analyze a game with multiple players and common shocks.
method Probabilistic approach to study the game, including mean field and FBSDEs.
result Unique equilibrium found for both N-player and mean field games.
Kernel methods are studied in a mean field limit for high-dimensional data.
problem Analyzing kernel methods in high-dimensional data with many variables.
method Investigation of kernel methods in the mean field limit of interacting particle systems.
result Rigorous mean field limit of kernels and detailed analysis of the limiting reproducing kernel Hilbert space.
Study analyzes adversarial training dynamics without data distribution assumptions.
problem Understanding training dynamics of adversarial training without data distribution assumptions.
method Mean field theory approach to analyze adversarial training in random deep neural networks.
result Upper bounds of adversarial loss derived empirically and theoretically.
A new method uses Mean Field Games to optimize mixture models of Bernoulli and categorical distributions.
problem Optimizing parameters of finite mixture models of Bernoulli and categorical distributions.
method Mean Field Games theory applied to multi-population systems.
result The Mean Field Games approach provides a method to compute mixture model parameters.
When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory lang…
In his lectures at College de France, P.L. Lions introduced the concept of Master equation, see [5] for Mean Field Games. It is introduced in a heuristic fashion, from the system of partial differential equations, associated to a Nash equilibrium for a large, but finite, number of players. The method, also explained in…
Paper studies a generalized mean field equation on closed Riemann surfaces.
problem Existence of solutions to a generalized mean field equation on closed Riemann surfaces.
method Uniform bound derivation and Leray-Schauder degree theory, minimax method.
result Existence results for solutions when α<λ1(Σ). Generalizes Carathéodory form for higher-order field theories.
problem Extending the Carathéodory form to second and higher-order Lagrangians.
method Geometric operations applied to the Poincaré--Cartan form and Lepage forms.
result Generalized Carathéodory form for second and higher-order Lagrangians.
A recent paper by Moore and Witten explained that Ramond-Ramond fields in Type II superstring theory have a global meaning in K-theory. In this note we amplify and generalize some points raised in that paper. In particular, we express the coupling of the Ramond-Ramond fields to D-branes in a K-theoretic framework and s…
The Habiro ring of a number field uses power series to study algebraic K-theory.
problem Analyzing algebraic K-theory of number fields.
method Introduces Habiro ring and modules graded by K3(K). result Establishes connections between Habiro ring and Chern-Simons theory.
Study on price formation among investors with exponential utility and liabilities.
problem Equilibrium price formation among investors with heterogeneous risk-averseness and liabilities.
method Mean-field game theory and mean-field backward stochastic differential equations (BSDE).
result Existence of equilibrium risk-premium process and market clearing in the large population limit.
Proves equations for high-dimensional gradient-based methods from Gaussian data.
problem High-dimensional asymptotics of gradient-based learning algorithms.
method Closed-form equations derived from dynamical mean-field theory.
result Equations match those from discretized DMFT for gradient flow.
As first noted in Korevaar, Kusner and Solomon ("KKS"), constant mean curvature implies a homological conservation law for hypersurfaces in ambient spaces with Killing fields.In Theorem 3.5 here, we generalize that law by relaxing the topological restrictions assumed in [KKS] and by allowing a weighted mean curvature f…
Mean field game theory studies the behavior of a large number of interacting individuals in a game theoretic setting and has received a lot of attention in the past decade (Lasry and Lions, Japanese journal of mathematics, 2007). In this work, we derive mean field game partial differential equation systems from determi…
In this paper, we develop a general framework of geometric functorial field theories, meaning that all bordisms in question are endowed with geometric structures. We take particular care to establish a notion of smooth variation of such geometric structures, so that it makes sense to require the output of our field the…
The paper analyzes the mean field Langevin dynamics and its convergence rate.
problem The convergence property of the mean field Langevin dynamics in the context of neural networks.
method The analysis uses a proximal Gibbs distribution and techniques from convex optimization.
result A concise convergence rate analysis of the mean field Langevin dynamics in both continuous and discrete time settings.
Model asset pricing with habit formation in a large market.
problem Understanding asset pricing in large heterogeneous markets with habit formation.
method Mean field game theory and quadratic-growth mean field BSDEs.
result Derives a semi-analytic solution for asset pricing model.
Deep learning enhances solving complex mean field games in finance.
problem Solving large-scale mean field games with financial applications.
method Combining mean field games theory with deep learning techniques.
result Improved solutions for large-scale financial games.
Recent studies have suggested that the cognitive process of the human brain is realized as probabilistic inference and can be further modeled by probabilistic graphical models like Markov random fields. Nevertheless, it remains unclear how probabilistic inference can be implemented by a network of spiking neurons in th…
Theory for soft-margin classifiers on object manifolds.
problem Classifying object manifolds with variability.
method Mean-field theory of soft-margin classifiers applied to object manifolds.
result Prediction of classification errors and their dependence on regularization.
Study on LOB dynamics using mean-field game theory.
problem Modeling liquidity dynamics in limit order books.
method Mean-field stochastic differential equation and control problem formulation.
result Equilibrium density function of LOB can be derived.
Unified analysis of DLNs using DMFT reveals dynamics of loss convergence and generalization trade-offs.
problem Understanding the overall dynamics of diagonal linear networks (DLNs) in neural network training.
method Dynamical Mean-Field Theory (DMFT) applied to DLNs.
result Derives low-dimensional effective process capturing high-dimensional gradient flow dynamics.
Game theory models how agents trade in a risky asset considering price impact and a common signal.
problem Modeling how financial agents liquidate assets in a risky market with price impact and a common signal.
method Formulated and solved a multi-player stochastic differential game and mean field game.
result Equilibrium strategies reveal how agents adjust the predictive trading signal to price impact.
The paper explores polysymplectic structures and their reductions in field theories.
problem Invariance of Lagrangian and Hamiltonian field theories under symmetry groups.
method Application of polysymplectic reduction theorem for both Lagrangian and Hamiltonian field equations.
result Identification and relation of polysymplectic structures through Routhian function and Legendre transformation.
Efficiently reduces rank of non-negative matrices with quadratic time complexity.
problem Efficiently reducing the rank of non-negative matrices.
method Formulated rank reduction as a mean-field approximation using a log-linear model.
result Optimal solution for minimizing KL divergence can be computed in closed form.
Massless scalar and vector fields are coupled to Lyra geometry by means of Duffin-Kemmer-Petiau (DKP) theory. Using Schwinger Variational Principle, equations of motion, conservation laws and gauge symmetry are implemented. We find that the scalar field couples to the anholonomic part of the torsion tensor, and the gau…