We propose two fast covariance smoothing methods and associated software that scale up linearly with the number of observations per function. Most available methods and software cannot smooth covariance matrices of dimension \(J>500\); a recently introduced sandwich smoother is an exception but is not adapted to smooth covariance matrices of large dimensions, such as \(J= 10{,}000\). We introduce two new methods that circumvent those problems: (1) a fast implementation of the sandwich smoother for covariance smoothing; and (2) a two-step procedure that first obtains the singular value decomposition of the data matrix and then smoothes the eigenvectors. These new approaches are at least an order of magnitude faster in high dimensions and drastically reduce computer memory requirements. The new approaches provide instantaneous (a few seconds) smoothing for matrices of dimension \(J=10{,}000\) and very fast (\(<\)10 min) smoothing for \(J=100{,}000\). R functions, simulations, and data analysis provide ready to use, reproducible, and scalable tools for practical data analysis of noisy high-dimensional functional data.

中文翻译：

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高维函数数据的快速协方差估计。

我们提出了两种快速协方差平滑方法和相关软件，它们随着每个函数的观测数量线性扩展。大多数可用的方法和软件无法平滑维数为\(J>500\)的协方差矩阵；最近推出的三明治平滑器是一个例外，但它不适合平滑大维度的协方差矩阵，例如\(J= 10{,}000\)。我们引入了两种新方法来规避这些问题：（1）快速实现用于协方差平滑的三明治平滑器；(2) 两步过程，首先获得数据矩阵的奇异值分解，然后平滑特征向量。这些新方法在高维度上速度至少快一个数量级，并大大降低了计算机内存需求。新方法为维度为\(J=10{,}000\)的矩阵提供瞬时（几秒）平滑，并为\(J=100{,}000\ ) 提供非常快速（\(<\) 10 分钟）的平滑）。R 函数、模拟和数据分析为噪声高维函数数据的实际数据分析提供了即用型、可重复且可扩展的工具。