Sequential and Simultaneous Distance-based Dimension Reduction

This paper introduces a method called Sequential and Simultaneous Distance-based Dimension Reduction (\(S^2D^2R\)) that performs simultaneous dimension reduction for a pair of random vectors based on Distance Covariance (dCov). Compared with Sufficient Dimension Reduction (SDR) and Canonical Correlation Analysis (CCA)-based approaches, \(S^2D^2R\) is a model-free approach that does not impose dimensional or distributional restrictions on variables and is more sensitive to nonlinear relationships. Theoretically, we establish a non-asymptotic error bound to guarantee the performance of \(S^2D^2R\). Numerically, \(S^2D^2R\) performs comparable to or better than other state-of-the-art algorithms and is computationally faster. All codes of our \(S^2D^2R\) method can be found on Github, including an R package named S2D2R.