Conditional Uncorrelation and Efficient Non-approximate Subset Selection in Sparse Regression

TL;DR

This paper introduces a novel non-approximate subset selection method for sparse regression based on conditional uncorrelation, significantly reducing computational complexity and enhancing efficiency in high-dimensional data analysis.

Contribution

It proposes a new formula for conditional uncorrelation and an efficient subset selection method that avoids regression coefficient calculations, improving computational speed.

Findings

Reduces computational complexity from O(1/6 k^3 + mk^2 + mkd) to O(1/6 k^3 + 1/2 mk^2).

Enables faster subset selection in high-dimensional sparse regression.

Provides a non-approximate approach suitable for large datasets.

Abstract

Given $m$ $d$-dimensional responsors and $n$ $d$-dimensional predictors, sparse regression finds at most $k$ predictors for each responsor for linear approximation, $1\leq k \leq d-1$. The key problem in sparse regression is subset selection, which usually suffers from high computational cost. Recent years, many improved approximate methods of subset selection have been published. However, less attention has been paid on the non-approximate method of subset selection, which is very necessary for many questions in data analysis. Here we consider sparse regression from the view of correlation, and propose the formula of conditional uncorrelation. Then an efficient non-approximate method of subset selection is proposed in which we do not need to calculate any coefficients in regression equation for candidate predictors. By the proposed method, the computational complexity is reduced from $O(\frac{1}{6}{k^3}+mk^2+mkd)$ to $O(\frac{1}{6}{k^3}+\frac{1}{2}mk^2)$ for each candidate subset in sparse regression. Because the dimension $d$ is generally the number of observations or experiments and large enough, the proposed method can greatly improve the efficiency of non-approximate subset selection.