Bi-Lipschitz Euclidean Embeddings of Metric Spaces induced by Finite
Group Representations

Speaker: Radu Balan, University of Maryland

Abstract: In this talk we survey a series of results for sorting-based
embeddings of orbits produced by finite group actions.
Consider a finite group G acting by isometries $g\mapsto U(g)$  on a finite
dimensional Euclidean space V. Given a fixed vector w ("window" or
"template"),
we study the sorted co-orbit map $x \mapsto T_w(x) =
\downarrow(<x,U(g)w>)_g$, where the entries are sorted in nonincreasing
order. The central problem is to construct a collection of windows
$w_1,...w_p$ such that the combined map $x\mapsto
(T_{w_1}(x),...,T_{w_p}(x))$ is a bi-Lipschitz embedding of the orbit space.
Our analysis shows that at most $2 dim(V)-d_G$ windows are required, where
$d_G$ is the number of linearly independent invariant vectors. In several
important cases this bounds can be improved further. This talk is based on
joint work with Nadav Dym, Efstratios Tsoukanis and Matthias Wellershoff.
See arXiv:2308.11784 , 2310.16365 , 2410.05446 , 2510:22186

December 5th, 2025

Reception at 3:30 PM in 306 Math Science Bldg

Talk at 4 PM in Strickland 114