Characterizes nodal volumes of Gaussian fields on manifolds, extending previous work.
problem Understanding the law and regularity of nodal volumes for Gaussian fields on manifolds.
method Gaussian measures, Morse theory, Malliavin-Sobolev spaces, ray absolute continuity.
result Extension and generalization of previous work on stationary fields to arbitrary dimensions.
The abstract proposes a neural network theory using quantum field theory.
problem Understanding the behavior of neural networks in the asymptotic and non-asymptotic limits.
method Mapping neural networks to Wilsonian effective field theory, using Gaussian processes and Feynman diagrams.
result Established a direct connection between overparameterization and simplicity of neural network likelihoods.
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.
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.
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 finds Calabi-Yau models' operator spectra match random matrix theory.
problem Understanding spectra of Calabi-Yau sigma models.
method Numerical methods for Ricci-flat metrics, averaging over complex structure moduli space.
result Spectrum matches Gaussian orthogonal ensemble of random matrix theory.
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…
Research proves the semi-classical limit of Liouville conformal field theory, describing deterministic geometry from random fluctuations.
problem Proving the semi-classical limit of Liouville conformal field theory.
method Probabilistic definition of Liouville theory, proving existence of semi-classical limit, defining classical stress-energy tensor.
result Existence and description of the semi-classical limit in terms of a massive Gaussian free field with Robin boundary conditions.
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.
These notes were inspired by the course ''Quantum Field Theory from a Functional Integral Point of View'' given at the University of Zurich in Spring 2017 by Santosh Kandel. We describe Feynman's path integral approach to quantum mechanics and quantum field theory from a functional integral point of view, where the mai…
We use the explicit relation between genus filtrated s s s -loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich--Penner matrix model (KPMM), which is the generating function for volumes of discretized (open) moduli spaces M g , s d i s c M_{g,s}^{disc} M g , s d i sc (discrete volumes), to express Gaussian means…
CMRFs extend PGMs for topological data, capturing both conditional and marginal dependencies.
problem Limited expressiveness of PGMs for topological data.
method Introducing Colored Markov Random Fields (CMRFs) that model Gaussian edge variables on topological spaces.
result CMRFs improve distributed estimation over physical networks compared to baselines.
New GM layers improve neural network performance.
problem Improving neural network performance.
method Employing Gaussian mixture models and Wasserstein gradient flows.
result GM layers achieve comparable performance to two-layer networks.
The Heath-Jarrow-Morton (HJM) formulation of treasury bonds in terms of forward rates is recast as a problem in path integration. The HJM-model is generalized to the case where all the forward rates are allowed to fluctuate independently. The resulting theory is shown to be a two-dimensional Gaussian quantum field theo…
Study modular surfaces in Lorentz-Minkowski 3-space, classifying and analyzing their curvature and applications.
problem Understanding the curvature properties of modular surfaces in Lorentz-Minkowski space.
method Analyzing the sign of Gaussian and mean curvature, classifying surfaces, and applying to conformal field theories.
result Complete classification of zero Gaussian curvature modular surfaces and non-existence of non-planar maximal modular surfaces.
A stochastic theory for the toppling activity in sandpile models is developed, based on a simple mean-field assumption about the toppling process. The theory describes the process as an anti-persistent Gaussian walk, where the diffusion coefficient is proportional to the activity. It is formulated as a generalization o…
Generative models use kernel smoothing for conditioning on small example sets.
problem Improving generative models' performance with limited conditioning examples.
method Showed that cross-attention conditioning is equivalent to kernel smoothing, specifically a Nadaraya--Watson kernel smoother.
result The approach predicts and confirms three failure regimes for kernel-based conditioning.
Given a Gaussian Markov random field, we consider the problem of selecting a subset of variables to observe which minimizes the total expected squared prediction error of the unobserved variables. We first show that finding an exact solution is NP-hard even for a restricted class of Gaussian Markov random fields, calle…
Motivated by a sampling problem basic to computational statistical inference, we develop a nearly optimal algorithm for a fundamental problem in spectral graph theory and numerical analysis. Given an n × n n\times n n × n SDDM matrix M {\bf \mathbf{M}} M , and a constant − 1 ≤ p ≤ 1 -1 \leq p \leq 1 − 1 ≤ p ≤ 1 , our algorithm gives efficient access to a…
We analyze training dynamics in Gaussian mixture models using a comparison theorem.
problem Analyzing training algorithms with Gaussian mixture data.
method Applying a Gaussian comparison theorem to a specific family of training algorithms.
result Validated dynamic mean-field expressions and provided iterative refinement schemes.
New theory predicts deep neural networks can operate in an extended critical regime without fine-tuning.
problem Understanding the dynamics and computational principles of deep neural networks.
method Combining theories of heavy-tailed random matrices and non-equilibrium statistical physics.
result Deep neural networks can operate in an extended critical regime without fine-tuning parameters.
NIFTy.re accelerates imaging models and expands Gaussian processes and variational inference.
problem Slow performance and limited inference strategies in NIFTy.
method Rewritten NIFTy with new modeling principles, inference strategies, and JAX integration.
result Dramatic acceleration of models and new inference capabilities.
New theory predicts deep neural network learning curves.
problem Understanding and optimizing deep neural networks.
method Gaussian field theory and renormalization group.
result Accurate predictions of deep neural network learning curves.
In the framework of the supervised learning of a real function defined on a space X , the so called Kriging method stands on a real Gaussian field defined on X. The Euclidean case is well known and has been widely studied. In this paper, we explore the less classical case where X is the non commutative finite group of …
A new method for uncertainty estimation in neural networks using Gaussian-softmax integration.
problem Quantifying uncertainty in neural network predictions.
method Proposes a single-model approach integrating Gaussian distribution with softmax outputs, using mean-field approximation.
result Competitive performance on uncertainty estimation tasks and outperforms many methods on out-of-distribution detection.
We analyze quantum Yang-Mills theory on R 2 \mathbb{R}^2 R 2 using a novel discretization method based on an algebraic analogue of stochastic calculus. Such an analogue involves working with "Gaussian" free fields whose covariance matrix is indefinite rather than positive definite. Specifically, we work with Lie-algebra valu…
Gaussian processes improved for ocean current reconstruction and divergence identification.
problem Reconstructing ocean currents from sparse buoy data.
method Proposed a Helmholtz decomposition-based approach to Gaussian processes for better physical modeling.
result Improved inference on ocean currents and divergence identification with minimal computational cost.
This paper improves the neural network-QFT correspondence by nonperturbative renormalization.
problem Understanding neural networks through effective field theory and renormalization.
method Improves Wilsonian renormalization using nonperturbative renormalization group analysis.
result Changing standard deviation in neural networks can be interpreted as a renormalization flow.
A scalable factorized Gaussian process VAE for faster inference.
problem Inference bottlenecks in Gaussian process VAEs.
method Factorizes latent kernel across auxiliary features, leveraging independence.
result Significant speed-up in inference time (in theory and practice).
New framework models neural systems with random architecture on manifolds.
problem Complex, uncertain systems with non-Gaussian outputs.
method Latent random field on compact manifold generates neural architecture and weights.
result Synthetic neural systems can produce stochastic outputs for deterministic inputs.
The study proves the finiteness of moments for Gaussian field zeros and critical points.
problem Finiteness of moments for Gaussian field zeros and critical points.
method Definition and study of multijets, construction of p-multijet bundles.
result Linear statistics of Gaussian field zeros have finite p-th moments for p ≥ 1.
Study on Gaussian random fields' singularities on manifolds.
problem Understanding singularities of Gaussian random fields on manifolds.
method Computed expected values of singularities under various conditions.
result Explicit formulae for singularities under different constraints.
New model improves field learning with improved equivariance.
problem Learning equivariant stochastic fields.
method Equivariant Gaussian processes and Steerable Conditional Neural Processes.
result SteerCNPs significantly improve performance in transfer learning tasks.
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 methods reduce computational cost for Gaussian Markov Random Fields with sparse constraints.
problem Inference and simulation of GMRFs are computationally prohibitive with many constraints.
method Proposes a basis transformation into blocks of constrained and non-constrained subspaces.
result Significantly outperforms existing alternatives in computational cost.
The paper introduces novel Gaussian process models for vector-valued signals on manifolds.
problem Modeling vector-valued signals on non-Euclidean domains, especially for applications like wind speeds.
method Intrinsically defined Gaussian vector fields on manifolds, accounting for manifold geometry.
result Gaussian vector fields provide more refined inductive biases than extrinsic fields.
Paper calculates KL divergence for isotropic Gaussian-Markov fields.
problem Measuring divergence between isotropic Gaussian-Markov fields.
method Derives closed-form KL divergence expressions.
result Develops new similarity measures in image processing.
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.
The inference of correlated signal fields with unknown correlation structures is of high scientific and technological relevance, but poses significant conceptual and numerical challenges. To address these, we develop the correlated signal inference (CSI) algorithm within information field theory (IFT) and discuss its n…
Random Gaussian fields on 4D Riemannian manifolds with conformal invariance.
problem Characterizing and analyzing Gaussian fields on 4D Riemannian manifolds.
method Constructing and analyzing co-biharmonic Gaussian fields with covariance kernels defined by the Paneitz operator.
result Rigorous derivation of quantum Liouville measure for ∣ γ ∣ < 8 |γ|<\sqrt8 ∣ γ ∣ < 8 . Paper introduces Tensor Gauge Flow Models for better data encoding.
problem Lack of expressive flow dynamics in existing Generative Flow Models.
method Incorporates higher-order Tensor Gauge Fields into the Flow Equation.
result Tensor Gauge Flow Models achieve improved generative performance.
We focus on the problem of estimating and quantifying uncertainties on the excursion set of a function under a limited evaluation budget. We adopt a Bayesian approach where the objective function is assumed to be a realization of a Gaussian random field. In this setting, the posterior distribution on the objective func…
New control methods improve dynamic measure transport paths.
problem Improving paths for dynamic measure transport.
method Connecting mean-field games to optimization problems for learning paths, advocating for smoothness of velocities.
result Our method recovers more efficient and smooth transport models compared to untilted paths.
We prove combinatorially the explicit relation between genus filtrated s s s -loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich--Penner matrix model (KPMM). The latter is the generating function for volumes of discretized (open) moduli spaces M g , s d i s c M_{g,s}^{\mathrm{disc}} M g , s disc given by $N_{…
Constructs QFT on curved surfaces, proving axioms and calculating entropy.
problem Quantum Field Theory on curved surfaces and entanglement entropy.
method Local regularization, spectral truncation, gluing surfaces, CFT correlation functions, zeta determinants.
result Rigorously derived entropy calculation and geometric proofs.
New test detects sparse alternatives in Gaussian random fields.
problem Detecting sparse alternatives in Gaussian random fields.
method Ad-hoc Kac Rice formula for second maximum distribution, exact spacing test.
result Exact t t t -spacing test for high power in detecting sparse alternatives. Derives a new first order differential equation for smooth surfaces.
problem Finding new equations to describe smooth surfaces.
method Derives a linear differential equation of the first order.
result Proves the maximum principle for Darboux rotation fields.
The paper proves that Gaussian field critical points have finite moments.
problem Proving the finiteness of moments for Gaussian field critical points.
method General approach not specific to critical points, using Taylor polynomial non-degeneracy.
result The finiteness of moments of the number of critical points of Gaussian fields.