The Lie group SU(N) is very important in fundamental particle physics as a Gauge symmetry group. Building SU(N) Gauge-equivariant models is key to make ML useful in this field. It was not easy, but WE DID IT!
SU(N) gauge equivariant flows for LatticeQCD:
arxiv.org/abs/2008.05456
SU(N) gauge equivariant flows for LatticeQCD:
arxiv.org/abs/2008.05456
This work is the result of the most amazing collaboration between @MIT, @nyuniversity and @DeepMind and I couldn't be more thankful and proud to have the opportunity to work with this team.
The team: Denis Boyda, Gurtej Kanwar, @sracaniere,@DaniloJRezende,@msalbergo,@kylecranm, Daniel Hackett, Phiala Shanahan 
In Lattice-QCD we want to compute expectations under a path-integral with paths weighted by exp(-S(U)), where U \in SU(N) are the Gauge fields. And S(U) is a Gauge-invariant functional of U called "action".
What our work is addressing is this: How to build and learn a good proposal density m(U) that is easy to sample from while respecting the Gauge-symmetries of S(U)?
In our previous work arxiv.org/abs/1904.12072 we sorted this out for the Gauge group U(1), which is an Abelian group.
In our previous work arxiv.org/abs/1904.12072 we sorted this out for the Gauge group U(1), which is an Abelian group.
But Abelian (commutative) groups are much simpler to work with then non-Abelian groups such as SU(N).
On top of that SU(N) itself gets more and more complicated as we grow N (see thread below - this gets pretty cool! - as an appetiser enjoy the Weyl chambers of SU(4) gif)
On top of that SU(N) itself gets more and more complicated as we grow N (see thread below - this gets pretty cool! - as an appetiser enjoy the Weyl chambers of SU(4) gif)
To build a flow that defines a Gauge-invariant density we need two things: (1) An invariant base density and (2) A Gauge-equivariant flow.
1) Is easy, just start from the Haar measure in SU(N)
But what about (2)? ... It took us several months to figure this out.
1) Is easy, just start from the Haar measure in SU(N)
But what about (2)? ... It took us several months to figure this out.
We show that to build a SU(N) Gauge equivariant flow, we can first build a SU(N) conjugation equivariant flow. This dramatically simplifies the problem! Since Gauge transformations involve neighbouring links in the lattice whereas conjugation happens at a single point.
This lead us to one of our major contributions: We demonstrated that any conjugation-equivariant flow is a "spectral flow", i.e., a flow operates on the set of eigenvalues of a complex matrix (aka its spectrum).
So we invented "spectral normalizing flows". The eigenvalues of an SU(N) matrix live inside a (N-1)-Torus called maximal Torus T. Since spectral flows operate on the spectrum, they must both live inside T and be permutation-equivariant with respect to eigenvalue permutations.
The restriction of the permutation group to the maximal torus is known as the Weyl group. So now the problem has been reduced to building Weyl-equivariant normalizing flow inside a (N-1)-Torus. Bored? Just keep going :)
The maximal torus can be broken down into N! "cells" where the Weyl group acts transitively and freely. Any Weyl-equivariant density on the torus T is then made of N! copies of the density on a single cell. We call flows on this space "Maximal Torus Flows".
A way to build such flow is to build a "canonicalization map" which projects every cell into a "canonical" cell, then build the flow on the canonical cell and undo the projection. Attached we show the cell structure and canonical cell (orange) for N=2,3,4.
But, how to build flows on cells? It is known from group theory that these cells are the exp map of (N-1)-simplexes living in a subspace of the tangent space of known as Cartan sub-algebra. Based on this theory, we show how to compute explicitly the vertices of this simplex.
This is when we introduce another contribution of this work: N-simplex flows. We show how one can build normalizing flows on arbitrary N-simplexes via a sequence on function compositions illustrated in the figure for the 2-simplex case.
Putting everything together, we've built this behemoth of flows inside flows inside flows shown in the figure.
But does it work? YES
That was a lot of work! It is an enormous pleasure and challenge to work on this problem. Thanks to everyone that made this possible!
To finish, enjoy a 3D projection of the 120 Weyl chambers of SU(5)
To finish, enjoy a 3D projection of the 120 Weyl chambers of SU(5)
@TacoCohen @wellingmax @ylecun @jhhalverson @KrippendorfSven I hope that you will enjoy reading this paper !
Also checkout thread by @KyleCranmer
@threadreaderapp unroll
Unrolled thread:
Sorry @KyleCranmer I cut your handle above by mistake.
