具有预测变量间图结构的稀疏Fréchet充分降维
Sparse Fréchet Sufficient Dimension Reduction with Graphical Structure Among Predictors
AI总结：
本文提出一种利用预测变量间图结构的高维稀疏Fréchet充分降维方法，通过构建避免协方差矩阵求逆的凸优化问题并采用ADMM算法求解，实现了子空间估计与变量选择一致性。
摘要：
Fréchet回归在建模度量空间取值的复杂数据响应（如概率分布和单位球面上的向量）方面受到广泛关注。然而，现有文献多聚焦于预测变量维度固定且样本量趋于无穷的经典设定。本文提出了一种针对高维欧几里得预测变量的稀疏Fréchet充分降维方法，该方法利用了预测变量间的图结构信息。具体而言，我们构建了一个凸优化问题，利用预测变量间的图信息，并避免了高维协方差矩阵的求逆运算。同时，我们提供了交替方向乘子法（ADMM）算法来求解该优化问题。理论上，在适当条件下，所提方法能实现子空间估计和变量选择的一致性。此外，通过大量模拟和实际数据分析，展示了该方法的有限样本性能。
英文摘要：
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.
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