Abstract:

Supervised dimension reduction has long been an important topic in statistics and machine learning. Sufficient dimension reduction (SDR) combines the idea of dimension reduction with the concept of sufficiency and it places the supervised dimension reduction in a rigorous statistical framework. In estimating the central subspace (CS), inverse regression based SDR methods involve solving a generalized eigenvalue problem. Solving such a problem can be problematic when the dimension of covariates, p, is much larger than the sample size n. In recent years, there are emerging new techniques in numerical linear algebra, called randomized algorithms or random sketching, for high dimensional and large scale problems. To overcome the large-p-small-n problem in SDR, we combine the idea of statistical inference with random sketching to propose a new SDR method, named integrated random-partition SDR (iRP-SDR). The method of iRP-SDR consists of (1) repeated sketches for the CS and (2) an integration of these multiple sketches to form a final estimate of the CS. The integration of multiple sketches involves taking weighted sample mean on a matrix Stiefel manifold. A geometry algorithm will be introduced. (This talk is based on joint works with Hung Hung, Rung-Sheng Lu, Hung Chen, National Taiwan University)