# Sequential and Simultaneous Distance-based Dimension Reduction

## TL;DR

This paper presents $S^2D^2R$, a model-free, nonlinear-sensitive dimension reduction method for pairs of random vectors based on Distance Covariance, with theoretical guarantees and superior computational performance.

## Contribution

Introduces $S^2D^2R$, a novel nonlinear, model-free dimension reduction technique with theoretical error bounds and improved computational efficiency.

## Key findings

- Performs comparably or better than existing methods
- Provides non-asymptotic error bounds
- Faster computational performance

## Abstract

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.