# Codes and Expansions (CodEx) Seminar

## Radu Balan (University of Maryland)

Embeddings of Metric Spaces induced by Permutation Groups

In this talk we discuss two actions of the symmetric group on Euclidean spaces. These actions induce metric structures on quotient spaces. Our problem is to analyze Lipschitz and bi-Lipschitz embeddings of these quotient spaces. The full Euclidean space is given by pairs \((A,X)\) composed of a symmetric matrix and a rectangular matrix with same number of rows, and the action is given by \((P;A,X) \rightarrow (PAP^T,PX)\). We analyze bi-Lipschitz representations of the metric space associated to \(A=0\). We discuss alsoinvariant representations associated to the \(X=0\) component. Constructions presented here are applied to two graph learning datasets: QM9, a quantum chemistry dataset, and Proteins, a biomolecular dataset. Two problems are considered: regression, on the QM9 dataset, and classification, on the Proteins dataset.