Introduction:
Physists has been suspecting the connection between machine learning and renormalization group for a long time. The best example would be CNN where the procedure resembles Coarse-Graining of Kadanoff. Recently, this connection has been put into an firm theoretical ground by Andrea Carosso, Jordan Cotler, Semon Rezchikov. In their works, they have established the map between Exact Renormalization Group equation and Stochastic Differential equation, which is the foundation of Diffusion model. Furthermore, they have reformulated renormalization group as an optimization problem.
In order to fully understand this connection, I am writing a note to go through every details. And I hope I can retell this whole story in a manner that is more friendly for people who are not familar with RG or at least I would attempt to do so.
- Wegner-Morris equation, Polchinski's equation
- Gradiant flow
- Intuitive introduction of Optimal transport problems
- Renormalization group flow recasted as an optimization problem
- Flow-matching's equivalence to Score-matching recasted as deterministic and stochastic formulation of RG