Non-Asymptotic Analysis Of Data Augmentation For Precision Matrix Estimation

Lucas Morisset, Adrien Hardy, Alain Oliviero Durmus

Keywords: Data Augmentation, Precision Matrix, High Dimensional Statistics, Random Matrix Theory

TL;DR: We introduce a data-centric estimator of the squared error of usual precision matrix estimates, both in the non-augmented case and the augmented case. Furthermore, we give non-asymptotic guarantees on our estimate.

Abstract:

This paper addresses the problem of inverse covariance (also known as precision matrix) estimation in high-dimensional settings. Specifically, we focus on two classes of estimators: linear shrinkage estimators with a target proportional to the identity matrix, and estimators derived from data augmentation (DA). Here, DA refers to the common practice of enriching a dataset with artificial samples—typically generated via a generative model or through random transformations of the original data—prior to model fitting.
For both classes of estimators, we derive estimators and provide concentration bounds for their quadratic error. This allows for both method comparison and hyperparameter tuning, such as selecting the optimal proportion of artificial samples.
On the technical side, our analysis relies on tools from random matrix theory. We introduce a novel deterministic equivalent for generalized resolvent matrices, accommodating dependent samples with specific structure. We support our theoretical results with numerical experiments.

Primary Area: Theory (e.g., control theory, learning theory, algorithmic game theory)

Submission Number: 13066