A unified generalization of the inverse regression methods via column selection

Abstract A bottleneck of sufficient dimension reduction (SDR) in the modern era is that, among numerous methods, only sliced inverse regression (SIR) is generally applicable in high-dimensional settings. The higher-order inverse regression methods, which form a major family of SDR methods superior to SIR at the population level, suffer from the dimensionality of their intermediate matrix-valued parameters which have excessive columns. In this paper, we propose to use a small subset of columns of the matrix-valued parameter for SDR estimation, which breaks the convention of using the ambient matrix in the higher-order inverse regression methods. With a quick column selection procedure, we then generalize these methods and their ensembles in high-dimensional sparse settings, in a uniform manner that resembles sparse SIR without additional assumptions. This is the first promising attempt in the literature to free the higher-order inverse regression methods from their dimensionality, thereby facilitating the application of SDR. Some numerical illustrations, including both simulation studies and a real data example, are provided at the end.