  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.
