Sparse Fréchet sufficient dimension reduction with graphical structure among predictors
针对高维欧几里得预测变量，提出一种利用图结构信息的稀疏弗雷歇充分降维方法，通过凸优化避免协方差矩阵求逆，实现子空间估计和变量选择一致性。
Abstract Fréchet 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 Fréchet regression literature focuses on the classical setting where the predictor dimension is fixed, and the sample size goes to infinity. This paper proposes sparse Fréchet 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.