Errors-in-variables Fr\'echet Regression with Low-rank Covariate Approximation

TL;DR

This paper introduces a novel low-rank Fréchet regression method that improves estimation accuracy for non-Euclidean responses with noisy and limited covariate data, combining theoretical analysis and empirical validation.

Contribution

It presents a new estimation framework that integrates low-rank covariate approximation with Fréchet regression, addressing high-dimensional and errors-in-variables challenges.

Findings

The proposed estimator has favorable large-sample properties.

Numerical experiments show improved performance over existing methods.

The method effectively handles noisy, high-dimensional covariate data.

Abstract

Fr\'echet regression has emerged as a promising approach for regression analysis involving non-Euclidean response variables. However, its practical applicability has been hindered by its reliance on ideal scenarios with abundant and noiseless covariate data. In this paper, we present a novel estimation method that tackles these limitations by leveraging the low-rank structure inherent in the covariate matrix. Our proposed framework combines the concepts of global Fr\'echet regression and principal component regression, aiming to improve the efficiency and accuracy of the regression estimator. By incorporating the low-rank structure, our method enables more effective modeling and estimation, particularly in high-dimensional and errors-in-variables regression settings. We provide a theoretical analysis of the proposed estimator's large-sample properties, including a comprehensive rate analysis of bias, variance, and additional variations due to measurement errors. Furthermore, our numerical experiments provide empirical evidence that supports the theoretical findings, demonstrating the superior performance of our approach. Overall, this work introduces a promising framework for regression analysis of non-Euclidean variables, effectively addressing the challenges associated with limited and noisy covariate data, with potential applications in diverse fields.