We propose a differential geometric approach for building families of low-rank covariance matrices, via interpolation on low-rank matrix manifolds. In contrast with standard parametric covariance classes, these families offer significant flexibility for problem-specific tailoring via the choice of “anchor” matrices for interpolation, for instance over a grid of relevant conditions describing the underlying stochastic process. The interpolation is computationally tractable in high dimensions, as it only involves manipulations of low-rank matrix factors. We also consider the problem of covariance identification, i.e., selecting the most representative member of the covariance family given a data set. In this setting, standard procedures such as maximum likelihood estimation are nontrivial because the covariance family is rank-deficient; we resolve this issue by casting the identification problem as distance minimization. We demonstrate the utility of these differential geometric families for interpolation and identification in a practical application: wind field covariance approximation for unmanned aerial vehicle navigation.

我们通过在低秩矩阵流形上进行插值，提出了一种用于构建低秩协方差矩阵族的微分几何方法。与标准参数协方差类相反，这些族通过选择“锚”矩阵进行插值，例如在描述潜在随机过程的相关条件网格上，为特定问题的剪裁提供了显着的灵活性。由于插值仅涉及对低阶矩阵因子的处理，因此该插值在高维范围内在计算上易于处理。我们还考虑了协方差识别的问题，即在给定数据集的情况下选择协方差家族中最具代表性的成员。在这种情况下，因为协方差序列是秩不足的，所以诸如最大似然估计之类的标准过程是不平凡的。我们通过将识别问题转换为距离最小化来解决此问题。我们演示了在实际应用中这些微分几何族在插值和识别中的实用性：用于无人机导航的风场协方差逼近。