Online Directional Regression for Streaming Sufficient Dimension Reduction

We develop an online directional regression for sufficient dimension reduction with streaming data. Unlike first-moment methods, directional regression exploits both inverse conditional means and inverse conditional variances, and can therefore recover central subspace directions generated by linear structures, symmetric dependencies, and interaction driven relationships. Extending directional regression to the online setting is nontrivial because its kernel depends on slice-wise inverse moments and on standardization by an unknown covariance matrix. We address these challenges by utilizing a stable slice probability free kernel, a ridge-stabilized recursive least-squares estimator of the standardization operator, metric-aware subspace tracking under a vanishing-ridge covariance metric, and a weighted-bootstrap ladle criterion for online structural dimension selection. Under standard regularity conditions, the online kernel estimator is root-$t$ consistent, the estimated generalized eigenspace consistently recovers the central subspace, and the proposed dimension selector consistently estimates the structural dimension. Simulation studies and real-data analyses demonstrate that the proposed method serves as a robust online alternative when the dominant data structure is unknown, adapting to both first-moment and second-moment-driven dependencies while yielding substantial computational savings relative to repeated batch directional regression.