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Non-Asymptotic Analysis of Data Augmentation for Precision Matrix Estimation

Lucas Morisset, Adrien Hardy, Alain Durmus

TL;DR

This work develops non-asymptotic random-matrix guarantees for precision-matrix estimation under two schemes: linear shrinkage and data augmentation. By introducing a deterministic equivalent for generalized resolvent matrices and deriving data-driven concentration bounds, it enables direct, data-based tuning of regularization and augmentation hyperparameters. The approach is validated with real-data experiments and supports practical guidance for choosing the amount and type of artificial data. Overall, the results illuminate how augmentation shapes the spectrum of the empirical covariance and improve stability in high-dimensional inverse covariance estimation.

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.

Non-Asymptotic Analysis of Data Augmentation for Precision Matrix Estimation

TL;DR

This work develops non-asymptotic random-matrix guarantees for precision-matrix estimation under two schemes: linear shrinkage and data augmentation. By introducing a deterministic equivalent for generalized resolvent matrices and deriving data-driven concentration bounds, it enables direct, data-based tuning of regularization and augmentation hyperparameters. The approach is validated with real-data experiments and supports practical guidance for choosing the amount and type of artificial data. Overall, the results illuminate how augmentation shapes the spectrum of the empirical covariance and improve stability in high-dimensional inverse covariance estimation.

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.

Paper Structure

This paper contains 22 sections, 21 theorems, 383 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and convention.
  3. Related work
  4. Inverse covariance estimation using shrinkage method
  5. Precision matrix estimation using generic data augmentation
  6. Numerical experiments
  7. Conclusion
  8. In-depth justification of the hypothesis
  9. Discussions on \ref{['ass:ProbaSmallEigenvalues']}
  10. Discussions on \ref{['ass:G_Lipschitz_Concentrated_cond_X']}, \ref{['ass:Smoothness']}, \ref{['ass:Stability']}
  11. Proof of \ref{['thm:NonAugmentedRidgeErrorEstimation']}
  12. Some sub Gaussian concentration bounds
  13. Concentration bounds for random quadratic forms
  14. A deterministic equivalent for $\operatorname{R}_X(\lambda)$, with arbitrarly small regularization parameter
  15. Conclusion on the proof of \ref{['thm:NonAugmentedRidgeErrorEstimation']}
  16. ...and 7 more sections

Key Result

Theorem 1

Assume ass:X_Lipschitz_Concentrated and ass:ProbaSmallEigenvalues. Then, it holds for all $t \geq 0$ and $\lambda >0$, for a universal constant $c>0$ and where Here $C_1, C_2 > 0$ are explicit polynomial functions of $\|\Sigma_X\|_\mathrm{op}^{-1}$, $\uplambda_d(\Sigma_X)$, $(\eta + \lambda)$ and $c_X^{-1}$, see Appendix B.

Figures (2)

  • Figure 1: Numerical results on MNIST for $\hat{\mathcal{E}}_X(\lambda)$ and $\hat{\mathcal{E}}_\mathrm{Aug}(\lambda)$, compared with \ref{['Proxies_def']}.
  • Figure 2: Numerical results on CIFAR-10 for $\hat{\mathcal{E}}_X(\lambda)$ and $\hat{\mathcal{E}}_\mathrm{Aug}(\lambda)$, compared with \ref{['Proxies_def']}.

Theorems & Definitions (38)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Remark 1
  • Theorem 2
  • Proposition 3
  • Theorem 3: Rudelson--Vershynin vershynin11
  • Corollary 1
  • proof
  • Definition 1: Lispchitz concentration
  • ...and 28 more