[Paper Review] On Principal Component Regression in a High-Dimensional Error-in-Variables Setting.

Anish Agarwal, Devavrat Shah+1 more|arXiv (Cornell University)|Oct 27, 2020

Statistical Methods and Inference30 references4 citations

TL;DR

This paper studies principal component regression (PCR) in high-dimensional error-in-variables models where covariates are corrupted by noise and missing values, and the number of covariates exceeds sample size. It establishes that PCR consistently estimates the minimum $β_2$-norm solution and provides non-asymptotic error rates for both estimation and out-of-sample prediction, even when out-of-sample covariates do not follow the same distribution as in-sample ones.

ABSTRACT

We analyze the classical method of principal component regression (PCR) in a high-dimensional error-in-variables setting. Here, the observed covariates are not only noisy and contain missing values, but the number of covariates can also exceed the sample size. Under suitable conditions, we establish that PCR identifies the unique linear model parameter with minimum $\ell_2$-norm, and derive non-asymptotic $\ell_2$-rates of convergence that show its consistency. Furthermore, we develop an algorithm for out-of-sample predictions in the presence of corrupted data that uses PCR as a key subroutine, and provide its non-asymptotic prediction performance guarantees. Notably, our results do not require the out-of-samples covariates to follow the same distribution as that of the in-sample covariates, but rather that they obey a simple linear algebraic constraint. We provide simulations that illustrate our theoretical results.

Motivation & Objective

• To understand the behavior of principal component regression (PCR) in high-dimensional settings with noisy and missing covariates.
• To establish theoretical consistency of PCR in estimating the minimum $β_2$-norm solution under high-dimensional error-in-variables models.
• To develop a prediction algorithm for out-of-sample data that remains valid even when the out-of-sample covariates do not follow the same distribution as in-sample covariates.
• To derive non-asymptotic $β_2$-norm error rates for both estimation and prediction under minimal distributional assumptions.

Proposed method

• The method analyzes PCR in a high-dimensional error-in-variables framework where observed covariates are corrupted by noise and may have missing values.
• It establishes that PCR identifies the unique minimum $β_2$-norm solution under suitable regularity conditions on the design matrix and error structure.
• The paper derives non-asymptotic $β_2$-norm convergence rates for the PCR estimator, showing consistency even when $p > n$.
• It proposes an out-of-sample prediction algorithm that uses PCR as a subroutine and provides performance guarantees under a linear algebraic constraint on the out-of-sample covariates.
• The theoretical analysis relies on high-dimensional random matrix theory and concentration inequalities to control estimation error.
• The method does not require the out-of-sample covariates to follow the same distribution as in-sample covariates, only that they satisfy a simple linear algebraic condition.

Experimental results

Research questions

• RQ1 Does PCR consistently estimate the minimum $β_2$-norm solution in high-dimensional error-in-variables models with noisy and missing covariates?
• RQ2 What non-asymptotic $β_2$-norm error rates can be derived for PCR estimation in high-dimensional settings with $p > n$?
• RQ3 Can PCR be reliably used for out-of-sample prediction when the out-of-sample covariates do not follow the same distribution as the in-sample covariates?
• RQ4 What conditions on the out-of-sample data ensure valid prediction performance when using PCR as a subroutine?
• RQ5 How do the theoretical error bounds for PCR depend on the noise level, dimensionality, and sample size in high-dimensional error-in-variables models?

Key findings

• PCR consistently estimates the unique minimum $β_2$-norm solution under suitable regularity conditions in high-dimensional error-in-variables models.
• Non-asymptotic $β_2$-norm error rates for PCR estimation are derived, showing consistency even when the number of covariates exceeds the sample size.
• The prediction algorithm based on PCR achieves non-asymptotic performance guarantees under a linear algebraic constraint on out-of-sample covariates, without requiring distributional similarity.
• The theoretical bounds do not depend on the out-of-sample covariates following the same distribution as in-sample covariates, only on a structural condition.
• Simulations confirm the theoretical predictions, demonstrating the robustness of PCR to noise, missing data, and high dimensionality.
• The results establish a theoretical foundation for using PCR in high-dimensional settings with corrupted data, extending its applicability beyond classical i.i.d. assumptions.