Motivated by establishing theoretical foundations for various manifold learning algorithms, we study the problem of Mahalanobis distance (MD), and the associated precision matrix, estimation from high-dimensional noisy data. By relying on recent transformative results in covariance matrix estimation, we demonstrate the sensitivity of \MD~and the associated precision matrix to measurement noise, determining the exact asymptotic signal-to-noise ratio at which MD fails, and quantifying its performance otherwise. In addition, for an appropriate loss function, we propose an asymptotically optimal shrinker, which is shown to be beneficial over the classical implementation of the MD, both analytically and in simulations. The result is extended to the manifold setup, where the nonlinear interaction between curvature and high-dimensional noise is taken care of. The developed solution is applied to study a multiscale reduction problem in the dynamical system analysis.

翻译：通过为多种多种学习算法建立理论基础,我们研究了Mahalanobis距离(MD)问题,以及相关的精确矩阵问题,从高维噪音数据中估算。我们借助最近对共变矩阵估计的变革结果,展示了\MD~和相关的精确矩阵对测量噪音的敏感性,确定了MD失败的准确的无线信号对噪音比率,并用其他方式量化了其性能。此外,为了适当的损失功能,我们提议了一种非现效的最佳缩水器,这证明有利于MDD的典型实施,包括分析和模拟。结果扩大到了多层结构,即曲线和高维度噪音之间的非线性互动得到了注意。开发的解决方案用于研究动态系统分析中的多尺度减少问题。