发表机构
Ghent University（根特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对可分希尔伯特空间上的共同协方差高斯随机元素，通过高阶矩算子刻画均值偏移下的高斯测度奇异性，推导了相关充要条件，解释了四阶谱量的概率起源并关联近完美分类 regime。
AI 中文摘要
对于可分希尔伯特空间上具有共同协方差算子的两个高斯随机元素，它们混合的四阶矩张量结构具有一种算子值表示，该表示由类条件二阶矩的纯二次分量得到。我们确定了协方差标准化后此四阶表示产生完全对角系数的张量机制，并证明这些系数完全由均值偏移的逐坐标Cameron-Martin能量控制。这为Fisher判别式成为诱导逐坐标算子的本征函数提供了一个充要条件及显式本征值。我们进一步针对两种互补构造引入了聚合四阶谱泛函，一种基于四阶矩构建，另一种基于期望乘积对应物构建，并证明当基础高斯测度相互奇异时，它们渐近等价。这些结果通过高阶矩算子提供了经典Cameron-Martin准则的谱实现，解释了先前为高斯判别和函数数据分类提出的四阶谱量的概率起源，并将它们与“近完美分类” regime关联起来。
英文摘要
For two Gaussian random elements on a separable Hilbert space with common covariance operator, the fourth-order moment tensor structure of their mixture admits an operator-valued representation obtained from the purely quadratic component of the class-conditional second moment. We identify the tensor mechanism generating the completely diagonal coefficients of this fourth-order representation after covariance standardization, and show that these coefficients are governed entirely by the coordinatewise Cameron-Martin energy of the mean shift. This yields a necessary and sufficient condition, together with an explicit eigenvalue, for the Fisher discriminant to be an eigenfunction of the induced coordinatewise operator. We further introduce aggregated fourth-order spectral functionals for two complementary constructions, one built from fourth-order moments and the other from a product-of-expectations counterpart, and prove that they are asymptotically equivalent precisely when the underlying Gaussian measures are mutually singular. These results provide a spectral realization of the classical Cameron-Martin criterion through higher-order moment operators, explaining the probabilistic origin of fourth-order spectral quantities previously proposed for Gaussian discrimination and functional data classification and relating them to the "near-perfect classification" regime.
Comments17 pages
