Optimal recovery of precision matrix for Mahalanobis distance from high-dimensional noisy observations in manifold learning

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