Riemannian Stochastic Optimization for Sufficient Dimension Reduction

Researchers have proposed two new algorithms, SMAVE and FlowSDR, for sufficient dimension reduction in high-dimensional regression, addressing limitations in existing methods.

Sufficient dimension reduction (SDR) is a technique that makes high-dimensional regression tractable by projecting covariates onto a low-dimensional subspace^[1]. Existing gradient-based estimators either suffer from the curse of dimensionality or have high computational costs. To address this, a team proposed SMAVE, which combines sparse projected-space nearest-neighbor localization with Riemannian stochastic gradient ascent^[1]. SMAVE has almost-sure convergence and a non-asymptotic rate matching standard non-convex stochastic first-order scaling. Around the same time, another research group introduced FlowSDR, a likelihood-based framework for SDR via conditional normalizing flows. FlowSDR jointly learns the projection and conditional density by maximizing a conditional log-likelihood^[2]. According to the researchers, FlowSDR is Fisher consistent under the SDR model and recovers the central subspace more accurately than existing SDR methods. Both papers were submitted to arXiv in May and June 2026, with the SMAVE paper submitted on an unspecified date in 2026 and the FlowSDR paper submitted on 31 May 2026^[1][2].