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.

翻译：Fr´echet回归已成为处理非欧几里得响应变量回归分析的有效方法。然而，其实际应用受限于对理想场景的依赖——即协变量数据丰富且无噪声。本文提出一种新的估计方法，通过利用协变量矩阵固有的低秩结构来克服这些限制。我们提出的框架融合了全局Fr´echet回归与主成分回归的概念，旨在提升回归估计器的效率与精度。通过引入低秩结构，本方法能够在高维及含测量误差的回归场景中实现更有效的建模与估计。我们对该估计量的大样本性质进行了理论分析，包括偏差、方差及测量误差导致的额外变异性的全面速率分析。此外，数值实验为理论结果提供了实证支持，验证了本方法的优越性能。总体而言，本文为非欧几里得变量的回归分析引入了一个有前景的框架，有效解决了有限且含噪声协变量数据的挑战，并在多个领域具有潜在应用价值。