Sufficient Dimension Reduction via Random-Partitions for Large-p-Small-n Problem

Abstract: Sufficient dimension reduction (SDR) is continuing an active research field nowadays for high dimensional data. It aims to estimate the central subspace (CS) without making distributional assumption. To overcome the large-$p$-small-$n$ problem we propose a new approach for SDR. Our method combines the following ideas for high dimensional data analysis: (1) Randomly partition the covariates into subsets and use distance correlation (DC) to construct a sketch of envelope subspace with low dimension. (2) Obtain a sketch of the CS by applying conventional SDR method within the constructed envelope subspace. (3) Repeat the above two steps for a few times and integrate these multiple sketches to form the final estimate of the CS. We name the proposed SDR procedure "integrated random-partition SDR (iRP-SDR)". Comparing with existing methods, iRP-SDR is less affected by the selection of tuning parameters. Moreover, the estimation procedure of iRP-SDR does not involve the determination of the structural dimension until at the last stage, which makes the method more robust in a high-dimensional setting. Asymptotic properties of iRP-SDR are also established. The advantageous performance of the proposed method is demonstrated via simulation studies and the EEG data analysis.