Distributed Mean Dimension Reduction Through Semi-parametric Approaches

In the present article we recast the semi-parametric mean dimension reduction approaches under a least squares framework, which turns the problem of recovering the central mean subspace into a series of problems of estimating slopes in linear regressions. It also facilitates to incorporate penalties to produce sparse solutions. We further adapt the semi-parametric mean dimension reduction approaches to distributed settings when massive data are scattered at various locations and cannot be aggregated or processed through 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 reduce the computational complexities of the pooled ones substantially. In addition, the distributed algorithms attain oracle rates after a finite number of iterations. We conduct extensive numerical studies to demonstrate the finite-sample performance of the distributed estimates and to compare with the pooled algorithms.