  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.
