机器学习的伽罗瓦定理：通过轻量级不变特征处理对称矩阵和点云的函数
A Galois theorem for machine learning: Functions on symmetric matrices and point clouds via lightweight invariant features
AI总结：
本研究借鉴伽罗瓦理论构造轻量级不变特征，提出对称矩阵（O(n²)特征）和固定维度点云（O(n)特征）的不变函数学习方法，结合DeepSets在分子性质回归、点云距离预测任务验证了可行性。
摘要：
本研究针对两类机器学习问题提出数学公式：（1）关于对称矩阵的函数，这类函数在置换共轭作用下保持不变；（2）关于点云的函数，这类函数在点的旋转、反射和置换作用下保持不变。为实现这一目标，我们借鉴伽罗瓦理论的思想，提出一种通用构造方法来生成可泛化分离的不变特征。我们构造了O(n²)个不变特征，这些特征源自n×n对称矩阵上有理函数域的生成元，且在行列联合置换下保持不变。研究表明，这些不变特征可分离对称矩阵的所有不同轨道，仅零测集除外；此类特征可用于对几乎所有加权图的不变函数进行通用逼近。对于固定维度的点云，我们证明不变特征的数量可减少至O(n)（n为点的数量），且泛化情况下不会损失表达能力。我们将这些不变特征与DeepSets结合，以学习不同大小的对称矩阵和点云的函数。通过分子性质回归和点云距离预测任务的实证研究，我们验证了该方法的可行性。
英文摘要：
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 provide a general construction of generically separating invariant features using ideas inspired by Galois theory. 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.
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