Galois features: Nearly-complete invariants on symmetric matrices

In this extended abstract, we summarize results from our recent work \citep{fullpaper}, in which we provide a mathematical formulation for learning functions on symmetric matrices that are invariant with respect to the action of permutations by conjugation. To achieve this, we construct $O(n^2)$ invariant features derived from generators for the field of rational functions on $n\times n$ symmetric matrices that are invariant under joint permutations of rows and columns. We obtain these generators using an argument from Galois theory. We show that these invariant features can separate all distinct orbits of symmetric matrices except for a measure zero set; such features can be used to universally approximate invariant functions on almost all weighted graphs. We empirically demonstrate the feasibility of our approach in a molecular properties regression problem.