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Implicit Riemannian Concave Potential Maps
Danilo Jimenez Rezende · Sébastien Racanière

Mon Dec 13 05:45 AM -- 06:00 AM (PST) @ None

We are interested in the challenging problem of modelling densities on Riemannian manifolds with a known symmetry group using normalising flows. This has many potential applications in physical sciences such as molecular dynamics and quantum simulations. In this work we combine ideas from implicit neural layers and optimal transport theory to propose a generalisation of existing work on exponential map flows, Implicit Riemannian Concave Potential Maps, IRCPMs. IRCPMs have some nice properties such as simplicity of incorporating knowledge about symmetries and are less expensive then ODE-flows. We provide an initial theoretical analysis of its properties and layout sufficient conditions for stable optimisation. Finally, we illustrate the properties of IRCPMs with density learning experiments on tori and spheres.

Author Information

Danilo Jimenez Rezende (Google DeepMind)
Sébastien Racanière (DeepMind)

Sébastien Racanière is a Staff Research Engineer in DeepMind. His current interests in ML revolve around the interaction between Physics and Machine Learning, with an emphasis on the use of symmetries. He got his PhD in pure mathematics from the Université Louis Pasteur, Strasbourg, in 2002, with co-supervisors Michèle Audin (Strasbourg) and Frances Kirwan (Oxford). This was followed by a two years Marie-Curie Individual Fellowship in Imperial College, London, and another postdoc in Cambridge (UK). His first job in the industry was at the Samsung European Research Institute, investigating the use of Learning Algorithms in mobile phones, followed by UGS, a Cambridge based company, working on a 3D search engine. He afterwards worked for Maxeler, in London, programming FPGAs. He then moved to Google, and finally DeepMind.

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