Stochastic quantization and diffusion models

Abstract

This is a pedagogical review of the possible connection between the stochastic quantization in physics and the diffusion models in machine learning. For machine-learning applications, the denoising diffusion model has been established as a successful technique, which is formulated in terms of the stochastic differential equation (SDE). In this review, we focus on an SDE approach used in the score-based generative modeling. Interestingly, the evolution of the probability distribution is equivalently described by a particular class of SDEs, and in a particular limit, the stochastic noises can be eliminated. Then, we turn to a similar mathematical formulation in quantum physics, that is, the stochastic quantization. We make a brief overview on the stochastic quantization using a simple toy model of the one-dimensional integration. The analogy between the diffusion model and the stochastic quantization is clearly seen in this concrete example. Finally, we discuss how the sign problem arises in the toy model with complex parameters. The origin of the difficulty is understood based on the Lefschetz thimble analysis. We point out that the SDE is not invariant under the variable change which induces a kernel and a special choice of the kernel guided by the Lefschetz thimble analysis can reduce the sign problem.

10 pages, 13 figures: Contribution to Special Topics in J.Phys.Soc.Jpn

References in corpus (20)

• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• The phase diagram of dense QCD
• High density QCD on a Lefschetz thimble?
• Lattice QCD at Finite Density -- An introductory review
• Universal features of JIMWLK and BK evolution at small x
• Hybrid Monte Carlo on Lefschetz Thimbles -- A study of the residual sign problem
• Finite-density lattice QCD and sign problem: current status and open problems
• Controlling complex Langevin dynamics at finite density
• The argument for justification of the complex Langevin method and the condition for correct convergence
• Introductory lectures on lattice QCD at nonzero baryon number
• Non linear gluon evolution in path-integral form
• Toward solving the sign problem with path optimization method
• Localised distributions and criteria for correctness in complex Langevin dynamics
• Application of neural network to sign problem via path optimization method
• Complex saddle points and the sign problem in complex Langevin simulation
• Hamilton dynamics for the Lefschetz thimble integration akin to the complex Langevin method
• Diffusion Models as Stochastic Quantization in Lattice Field Theory
• Restricted phase-space approximation in real-time stochastic quantization
• Analytical studies of the complex Langevin equation with a Gaussian Ansatz and multiple solutions in the unstable region
• Real-time correlators in 3+1D thermal lattice gauge theory

Cited by in corpus (2)

• Combining complex Langevin dynamics with score-based and energy-based diffusion models
• Efficient identification of critical regions via Flow Matching-based Monte Carlo initialization