Fr\'echet regression has received considerable attention to model metric-space valued responses that are complex and non-Euclidean data, such as probability distributions and vectors on the unit sphere. However, existing Fr\'echet regression literature focuses on the classical setting where the predictor dimension is fixed, and the sample size goes to infinity. This paper proposes sparse Fr\'echet sufficient dimension reduction with graphical structure among high-dimensional Euclidean predictors. In particular, we propose a convex optimization problem that leverages the graphical information among predictors and avoids inverting the high-dimensional covariance matrix. We also provide the Alternating Direction Method of Multipliers (ADMM) algorithm to solve the optimization problem. Theoretically, the proposed method achieves subspace estimation and variable selection consistency under suitable conditions. Extensive simulations and a real data analysis are carried out to illustrate the finite-sample performance of the proposed method.

翻译：弗雷歇回归在处理复杂非欧几里得数据（如概率分布和单位球面向量）的度量空间值响应方面受到广泛关注。然而，现有弗雷歇回归文献主要关注预测变量维度固定、样本量趋于无穷的经典设定。本文提出一种具有高维欧几里得预测变量间图结构的稀疏弗雷歇充分降维方法。具体而言，我们构建了一个利用预测变量间图信息并避免高维协方差矩阵求逆的凸优化问题，同时给出了交替方向乘子法（ADMM）算法求解该优化问题。理论上，所提方法在适当条件下实现了子空间估计与变量选择相合性。通过大量仿真实验和实际数据分析验证了所提方法的有限样本性能。