In this work, we present a mathematical formulation for machine learning of
(1) functions on symmetric matrices that are invariant with respect to the
action of permutations by conjugation, and (2) functions on point clouds that
are invariant with respect to rotations, reflections, and permutations of the
points. 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
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. For
point clouds in a fixed dimension, we prove that the number of invariant
features can be reduced, generically without losing expressivity, to $O(n)$,
where $n$ is the number of points. We combine these invariant features with
DeepSets to learn functions on symmetric matrices and point clouds with varying
sizes. We empirically demonstrate the feasibility of our approach on molecule
property regression and point cloud distance prediction.

提出了用于机器学习对称矩阵上不变函数和对点云上不变函数的数学公式，并结合 DeepSets 在不同尺寸的对称矩阵和点云上学习函数的可行性。