发表机构
QodeX Quantum, Inc.（QodeX量子公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究输入带李群作用的机器学习模型丢弃信息，定义衡量对称性的对象，通过牛顿迭代计算零纤维元素，成本低，开发相关应用并在分子性质预测等任务中测试，框架适用于经典神经网络和变分量子电路。
AI 中文摘要
我们为输入带有李群作用的机器学习模型所丢弃的信息开发了一个框架。给定李群\(G\)在空间\(V\)上的一个表示\(\pi\)以及一个学习函数\(f:V \to \mathbb{R}\)，我们定义了两个对象来衡量\(f\)不可见的对称性。在点\(x \in V\)处的零纤维是群元素的集合\(N_G(f,x) = \{g \in G: f(\pi(g^{-1}) \cdot x) = f(x)\}\)，其对\(x\)的逆作用无法被\(f\)检测到。当\(N_G(f,x)\)与\(x\)无关时，它与稳定子\(\mathrm{Stab}_G(f)\)重合，\(\mathrm{Stab}_G(f)\)是\(G\)中使\(f\)不变的最大子群。对于到\(\mathbb{R}\)的光滑映射，原像定理保证在一般输入下零纤维的维度至少为\(\dim G - 1\)，与架构无关。对于作用在自身上的紧群，彼得 - 外尔定理根据\(f\)的傅里叶系数矩阵给出了这两个对象的谱特征。我们表明零纤维元素可以通过在轨道映射上的牛顿迭代有效地计算，成本与几次梯度评估相当。在\(\mathrm{SO}(3)\)下的分子性质预测和\(\mathrm{PSL}(2, \mathbb{C})\)莫比乌斯群下的球面图像分类上，开发并通过实验测试了数据掩码、模型指纹识别和隐私保护计算的应用。该框架统一适用于经典神经网络和变分量子电路
英文摘要
We develop a framework for the information discarded by machine learning models whose inputs carry a Lie group action. Given a representation $π$ of a Lie group $G$ on a space $V$ and a learned function $f\colon V \to \mathbb{R}$, we define two objects measuring the symmetry invisible to $f$. The null fiber at a point $x \in V$ is the set $N_G(f,x) = \{g \in G : f(π(g^{-1}) \cdot x) = f(x)\}$ of group elements whose inverse action on $x$ is undetectable by $f$. When $N_G(f,x)$ is independent of $x$, it coincides with the stabilizer $\mathrm{Stab}_G(f)$, the largest subgroup of $G$ under which $f$ is invariant. For smooth maps to $\mathbb{R}$, the preimage theorem guarantees that null fibers have dimension at least $\dim G - 1$ at generic inputs, regardless of architecture. For compact groups acting on themselves, the Peter--Weyl theorem yields a spectral characterization of both objects in terms of the Fourier coefficient matrices of $f$. We show that null fiber elements can be computed efficiently via Newton iteration on the orbit map, at a cost comparable to a few gradient evaluations. Applications to data masking, model fingerprinting, and privacy-preserving computation are developed and tested experimentally on molecular property prediction under $\mathrm{SO}(3)$ and spherical image classification under the Möbius group $\mathrm{PSL}(2, \mathbb{C})$. The framework applies uniformly to classical neural networks and variational quantum circuits.
Comments22 pages, 10 figures
