Abstract: In this study we recast the semiparametric mean dimension reduction approaches under a least squares framework. This changes the problem of recovering the central mean subspace into a series of problems of estimating slopes in linear regressions, and enables us to incorporate penalties to produce sparse solutions. We further adapt the semiparametric mean dimension reduction approaches to distributed settings in which massive data are scattered at various locations, and cannot be aggregated or processed by a single machine. We propose three communication efficient distributed algorithms. The first yields a dense solution, the second produces a sparse estimation, and the third provides an orthonormal basis. The distributed algorithms are less complex computationally than a pooled algorithm, and attain oracle rates after a finite number of iterations. Using extensive numerical studies, we demonstrate the finite sample performance of the distributed estimates, and compare it with that of a pooled algorithm.

Key words and phrases: Central subspace, distributed estimation, sufficient dimension reduction.