Sparse Fréchet sufficient dimension reduction via nonconvex optimization

In the evolving landscape of statistical learning, exploiting low-dimensional structures, particularly for non-Euclidean objects, is an essential and ubiquitous task with wide applications ranging from image analysis to biomedical research. Among the momentous developments in the non-Euclidean domain, Fréchet regression extends beyond Riemannian manifolds to study complex random response objects in a metric space with Euclidean features. Our work focuses on sparse Fréchet dimension reduction where the number of features far exceeds the sample size. The goal is to achieve parsimonious models by identifying a low-dimensional and sparse representation of features through sufficient dimension reduction. To this end, we construct a multitask regression model with synthetic responses and achieve sparse estimation by leveraging the minimax concave penalty. Our approach not only sidesteps inverting a large covariance matrix but also mitigates estimation bias in feature selection. To tackle the nonconvex optimization challenge, we develop a double approximation shrinkage-thresholding algorithm that combines a linear approximation to the penalty term and a quadratic approximation to the loss function. The proposed algorithm is efficient as each iteration has a clear and explicit solution. Experimental results for both simulated and real-world data demonstrate the superior performance of the proposed method compared to existing alternatives.