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.

翻译：本文提出了一种用于机器学习以下两类函数问题的数学形式化方法：(1) 对称矩阵上关于共轭置换作用不变的函数；(2) 点云上关于旋转、反射及点排列变换不变的函数。为此，我们从 $n\times n$ 对称矩阵的行列联合置换不变有理函数域生成元中构造了 $O(n^2)$ 个不变特征。我们证明，这些不变特征能够区分除零测集以外的所有对称矩阵轨道；此类特征可用于在几乎所有加权图上通用逼近不变函数。对于固定维度的点云，我们证明在一般情况下，不变特征数量可降至 $O(n)$（$n$ 为点数）而不会损失表达能力。我们将这些不变特征与DeepSets相结合，以学习变长对称矩阵及点云上的函数。通过分子性质回归与点云距离预测任务，我们实证验证了该方法的可行性。