Normalizing Flows on Tori and Spheres ICML 2020 DeepMind - - PowerPoint PPT Presentation
Normalizing Flows on Tori and Spheres ICML 2020 DeepMind Collaborators Danilo Rezende George Papamakarios Sbastien Racanire Center for Cosmology and Particle Physics, NYU Center for Theoretical Physics, MIT Gurtej Kanwar Phiala
ICML 2020
Center for Cosmology and Particle Physics, NYU Center for Theoretical Physics, MIT DeepMind
Danilo Rezende George Papamakarios Sébastien Racanière Gurtej Kanwar Phiala Shanahan Michael Albergo Kyle Cranmer
Circles Tori Spheres
Angles Directions Joint configurations of molecules / robot arms
Wirnsberger & Ballard et al., Targeted free energy estimation via learned mappings, arxiv.org/abs/2002.04913, 2020
System of particles with periodic boundary conditions
Kanwar et al., Equivariant flow-based sampling for lattice gauge theory, arxiv.org/abs/2003.06413, 2020
Common techniques:
Density evaluation Sampling
Fix endpoints: Positive derivative: Match endpoint derivatives: Parameterize using angle:
Möbius transforms Circular splines (CS) Non-compact projections (NCP)
+ rotation to fix endpoints
Composing Möbius (or NCP) transformations does not increase expressivity since they form a group. Instead, we propose an efficient method to create mixtures of them. Assume
Still a valid diffeomorphism of S1 with tractable Jacobian!
Autoregressive flow whose conditionals are circle flows (Möbius, CS or NCP)
Comparison of Möbius, CS & NCP
Sphere to cylinder Cylinder to sphere
Target Flow samples Flow density
Target (top) vs flow (bottom) on a 3D sphere (shown are Mollweide projections)
Auto-reg Exp-map
Method
Autoregressive Exponential map
Pros
require any particular coordinate system)
sphere (no numerical instabilities)
symmetries
Cons
measure-zero from the n-sphere, this may lead to numerical issues
system
knowledge about density (e.g. symmetries)
○ As flexible as we like ○ Any-dimensional ○ Efficient and exact density evaluation and sampling