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Non-Asymptotic Analysis Of Data Augmentation For Precision Matrix Estimation
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Derives data-driven non-asymptotic concentration estimators for the quadratic error of ridge-type and data-augmented precision matrix estimators, using random matrix theory deterministic equivalents for dependent samples.
Problem
Estimating the precision matrix in high-dimensional regimes (d and n comparable) when the data is augmented with artificial samples (transformative or generative), where the quadratic error of the estimator is unknown and needs a computable proxy for comparing methods and tuning hyperparameters such
Approach
Uses random matrix theory to build a computable estimator of the quadratic error of the ridge-regularized inverse sample covariance and of the data-augmented inverse, with exponential concentration bounds; introduces a deterministic equivalent for generalized resolvents that handles dependence between original and artificial samples and covers vanishing regularization.
Key results
- On MNIST and CIFAR10 the data-driven error estimate hat E_X(lambda) closely tracks the empirical proxy and its minimizer matches that of the proxy across sample ratios gamma = d/n.
- Estimated augmented error hat E_Aug(0) is reported as a function of augmentation fraction alpha = m/(n+m) for Gaussian-mixture GDA and Gaussian-noise TDA, with minimizers matching the proxies.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Gaussian-mixture GDA (10 centroids) and Gaussian-noise TDA added to ridge-like estimator | precision matrix estimation (MNIST, CIFAR10) · MNIST and CIFAR10 | minimizer of estimated quadratic error vs proxy | No clear effect |
| Choosing lambda by minimizing the data-based estimate of the error | ridge-like precision estimator · MNIST and CIFAR10 | minimizer of estimated error vs proxy | Helped |
Limitations
- Third term of the error bound vanishes only when E[Lambda_G(X)] approximately commutes with Sigma_X
- Assumes sub-Gaussian / Lipschitz concentration of data and bounded augmentation distributions; generalization to finite moments left for future work
- Results rely on proxies using sample covariance of full dataset in experiments rather than exact errors
Experimental setup
Datasets: MNIST CIFAR10
Benchmarks: MNIST CIFAR10
Metrics: quadratic error of precision matrix estimator (Frobenius nor
Baselines: non-augmented ridge-like estimator
Builds on: Ledoit-Wolf shrinkage Couillet-Liao deterministic equivalents Lipschitz concentration
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