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Scalars are universal: Equivariant machine learning, structured like classical physics
Soledad Villar · David W Hogg · Kate Storey-Fisher · Weichi Yao · Ben Blum-Smith

Thu Dec 09 08:30 AM -- 10:00 AM (PST) @
There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincaré groups, at any dimensionality $d$. The key observation is that nonlinear O($d$)-equivariant (and related-group-equivariant) functions can be universally expressed in terms of a lightweight collection of scalars---scalar products and scalar contractions of the scalar, vector, and tensor inputs. We complement our theory with numerical examples that show that the scalar-based method is simple, efficient, and scalable.

Author Information

Soledad Villar (Johns Hopkins)
David W Hogg (New York University)

David W Hogg is Professor of Physics and Data Science at New York University and Group Leader of the Astronomical Data Group at the Flatiron Institute. His work is on computational data analysis in all areas of astrophysics, from extra-solar planet discovery to mapping the dark matter to measuring the expansion history of the Universe.

Kate Storey-Fisher (New York University)

Kate Storey-Fisher is a PhD candidate in Physics at New York University and a NASA FINESST Fellow. Her research is on the large-scale structure of the universe, focusing on statistical and data-science methods for observational cosmology.

Weichi Yao (New York University)
Ben Blum-Smith (NYU Center for Data Science)

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