Section: New Results

Slice Inverse Regression with Score Functions

Non-linear regression and related problems such as non-linear classification are core important tasks in machine learning and statistics. We consider the problem of dimension reduction in non-linear regression, which is often formulated as a non-convex optimization problem.

• We propose score function extensions to sliced inverse regression problems  [38],  [39], both for the first-order and second-order score functions, which provably improve estimation in the population case over the non-sliced versions; we study finite sample estimators and study their consistency given the exact score functions.

• We propose also to learn the score function as well (using score matching technique  [37]) in two steps, i.e., first learning the score function and then learning the effective dimension reduction space, or directly, by solving a convex optimization problem regularized by the nuclear norm.