This work studies finite sample approximations of the exact and entropic regularized Wasserstein distances between centered Gaussian processes and, more generally, covariance operators of functional random processes. We first show that these distances/divergences are fully represented by reproducing kernel Hilbert space (RKHS) covariance and cross-covariance operators associated with the corresponding covariance functions. Using this representation, we show that the Sinkhorn divergence between two centered Gaussian processes can be consistently and efficiently estimated from the divergence between their corresponding normalized finite-dimensional covariance matrices, or alternatively, their sample covariance operators. Consequently, this leads to a consistent and efficient algorithm for estimating the Sinkhorn divergence from finite samples generated by the two processes. For a fixed regularization parameter, the convergence rates are {\it dimension-independent} and of the same order as those for the Hilbert-Schmidt distance. If at least one of the RKHS is finite-dimensional, we obtain a {\it dimension-dependent} sample complexity for the exact Wasserstein distance between the Gaussian processes.

翻译：这份工作研究报告对中央高斯进程和一般的功能随机进程共变操作者之间准确和正常的瓦塞斯坦距离的有限抽样近似值进行了限定的抽样估计。 我们首先显示,这些距离/ 差异完全代表了复制核心Hilbert 空间( RKHS) 的共变和跨共变操作者, 与相应的共变函数相关。 使用此表示法, 我们显示, 两个中心高斯进程之间的辛克霍恩差异, 与其相应的标准化的有限共变和矩阵之间的差异, 或者与其样本共变操作者之间的差异相比, 可以一致和有效地估计。 因此, 这可以导致一种一致和高效的算法, 用于估算Sinkhorn与两个进程产生的有限样本的差异。 对于固定的正规化参数, 趋同率取决于 ~ lt 尺寸, 和与Hilbert- Schmidt 距离的顺序相同。 如果至少一个RKHS是定维, 我们获得高斯进程之间准确的瓦西列斯坦距离的样本复杂性。