Asymptotic properties of adaptive group Lasso for sparse reduced rank regression

TL;DR

This paper analyzes the asymptotic behavior of an adaptive group Lasso estimator in sparse reduced rank regression, demonstrating optimal convergence rates and variable selection consistency as dimensions grow.

Contribution

It establishes the asymptotic properties of the adaptive group Lasso estimator in high-dimensional reduced rank regression, including convergence rates and predictor selection consistency.

Findings

Achieves minimax optimal convergence rate.

Ensures variable selection consistency.

Handles growing dimensions with sample size.

Abstract

This paper studies the asymptotic properties of the penalized least squares estimator using an adaptive group Lasso penalty for the reduced rank regression. The group Lasso penalty is defined in the way that the regression coefficients corresponding to each predictor are treated as one group. It is shown that under certain regularity conditions, the estimator can achieve the minimax optimal rate of convergence. Moreover, the variable selection consistency can also be achieved, that is, the relevant predictors can be identified with probability approaching one. In the asymptotic theory, the number of response variables, the number of predictors, and the rank number are allowed to grow to infinity with the sample size.