Non-Asymptotic Analysis of Data Augmentation for Precision Matrix Estimation

2025-10-02

AI Summary

This paper addresses the problem of estimating high-dimensional inverse covariance (precision) matrices. To tackle structured dependence among samples—common in modern statistical learning—we propose a novel deterministic equivalent form for the generalized resolvent matrix, unifying linear shrinkage and data augmentation estimators. Our method yields the first non-asymptotically exact characterization of estimation error under data augmentation. Leveraging random matrix theory and generative data transformations, we derive tight, non-asymptotic quadratic error concentration bounds for both classes of estimators. Furthermore, we establish theoretically grounded hyperparameter tuning rules—e.g., for augmentation ratio—that balance bias and variance. All theoretical findings are rigorously validated through comprehensive numerical experiments, demonstrating substantial improvements in estimation stability and interpretability for high-dimensional precision matrices.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Data and user privacy-enhancing technologies for the Web

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.

Problem

• Analyzing precision matrix estimation in high-dimensional settings
• Deriving concentration bounds for data augmentation estimators
• Introducing deterministic equivalents for dependent sample analysis

Innovation

• Data augmentation for precision matrix estimation
• Novel deterministic equivalent for resolvent matrices
• Concentration bounds for quadratic error analysis