Koch-Janusz & Ringel: Mutual information, neural networks and the renormalization group

 

 

Journal Club 16.09.21

Claudia Merger

"Job" of RG: Keep important degrees of freedom (DOFs), integrate out the rest

"Job" of RG: Keep important degrees of freedom (DOFs), integrate out the rest

 

Difficulty: Which DOFs are important?

 

 

"Job" of RG: Keep important degrees of freedom (DOFs), integrate out the rest

 

Difficulty: Which DOFs are important?

 

Idea: Let a neural network (NN) find the answer for you

 

RSMI algorithm

  1. RSMI algorithm

  2. Application to Ising model

  3. Application to dimer model

  4. Discussion

RSMI algorithm

RSMI algorithm

maximize

RSMI algorithm

maximize

RSMI algorithm

Variant of "Information Bottleneck method"

3 restricted Boltzmann machines:

2 trained with contrastive divergence algorithm, learn            and        

 

3 restricted Boltzmann machines:

2 trained with contrastive divergence algorithm, learn            and        

Third learns              

3 restricted Boltzmann machines:

2 trained with contrastive divergence algorithm, learn            and        

Third learns              

using

3 restricted Boltzmann machines:

2 trained with contrastive divergence algorithm, learn            and        

Third learns              

using

to maximize

Then: Sample from            

and iterate the whole procedure

Application to Ising model

Application to Ising model

Application to Ising model

= Kardanoff

Block-spin

Application to Dimer model

Discussion

  • RSMI algorithm maximizes mutual information btw. input and hidden DOFs
  • Without prior knowledge, learns to perform RG coarse graining in two examples
  • Claim: Can be used to find relevant DOFs in cases where they are unknown

Mutual information

Goal: Train                on

 

have:         ,

 

 

Train:

 

 

use:

Estimating temperature