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

## TL;DR

This paper investigates the estimation of Mahalanobis distance and its precision matrix from noisy high-dimensional data, providing theoretical insights, optimal shrinkage methods, and applications to manifold learning and dynamical systems.

## Contribution

It introduces an asymptotically optimal shrinker for precision matrix estimation, extending results to manifold settings and analyzing the impact of noise on Mahalanobis distance.

## Key findings

- Identifies the noise threshold where Mahalanobis distance fails
- Proposes an optimal shrinker for precision matrix estimation
- Demonstrates improved performance over classical methods

## Abstract

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.