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Motivated by establishing theoretical foundations for various manifoldlearning algorithms, we study the problem of Mahalanobis distance (MD), and theassociated precision matrix, estimation from high-dimensional noisy data. Byrelying on recent transformative results in covariance matrix estimation, wedemonstrate the sensitivity of \MD~and the associated precision matrix tomeasurement noise, determining the exact asymptotic signal-to-noise ratio atwhich MD fails, and quantifying its performance otherwise. In addition, for anappropriate loss function, we propose an asymptotically optimal shrinker, whichis shown to be beneficial over the classical implementation of the MD, bothanalytically and in simulations. The result is extended to the manifold setup,where the nonlinear interaction between curvature and high-dimensional noise istaken care of. The developed solution is applied to study a multiscalereduction problem in the dynamical system analysis.

Optimal Recovery of Precision Matrix for Mahalanobis Distance from High Dimensional Noisy Observations in Manifold Learning