In this work we study two Riemannian distances between infinite-dimensional positive definite Hilbert-Schmidt operators, namely affine-invariant Riemannian and Log-Hilbert-Schmidt distances, in the context of covariance operators associated with functional stochastic processes, in particular Gaussian processes. Our first main results show that both distances converge in the Hilbert-Schmidt norm. Using concentration results for Hilbert space-valued random variables, we then show that both distances can be consistently and efficiently estimated from (i) sample covariance operators, (ii) finite, normalized covariance matrices, and (iii) finite samples generated by the given processes, all with dimension-independent convergence. Our theoretical analysis exploits extensively the methodology of reproducing kernel Hilbert space (RKHS) covariance and cross-covariance operators. The theoretical formulation is illustrated with numerical experiments on covariance operators of Gaussian processes.
翻译:在这项工作中,我们研究了与功能随机过程,特别是高森过程相关的共变操作员之间无限正向的Hilbert-Schmidt-Hilbert-Schmidt操作员与Log-Hilbert-Schmidt操作员之间的两处里曼尼的距离。我们的第一个主要结果显示,这两处距离都与Hilbert-Schmidt规范相融合。利用Hilbert空间估价随机变量的浓度结果,我们然后表明,从(一) 抽样共变操作员、(二) 定数、常态共变数矩阵和(三) 给定过程产生的定数样本中可以持续和有效地估算出两者的距离,这些样本都具有维独立的趋同。我们的理论分析广泛利用了再生产Hilbert空间(RKHS)常变数和交叉变数操作员的方法。关于高森过程共变数操作员的数字实验说明了理论的表述。