# A Selective Overview of Variable Selection in High Dimensional Feature Space (Invited Review Article)

## TL;DR

This review summarizes recent theoretical and methodological advances in high-dimensional variable selection, focusing on penalized likelihood methods, their properties, and applications to ultra-high dimensional data.

## Contribution

It provides a comprehensive overview of developments in penalized likelihood approaches, including non-concave penalties and ultra-high dimensional techniques like screening methods.

## Key findings

- Non-concave penalized likelihood methods have desirable statistical properties.
- Screening and two-scale methods are effective for ultra-high dimensional variable selection.
- Recent theory extends the applicability of variable selection techniques to higher dimensions.

## Abstract

High dimensional statistical problems arise from diverse fields of scientific research and technological development. Variable selection plays a pivotal role in contemporary statistical learning and scientific discoveries. The traditional idea of best subset selection methods, which can be regarded as a specific form of penalized likelihood, is computationally too expensive for many modern statistical applications. Other forms of penalized likelihood methods have been successfully developed over the last decade to cope with high dimensionality. They have been widely applied for simultaneously selecting important variables and estimating their effects in high dimensional statistical inference. In this article, we present a brief account of the recent developments of theory, methods, and implementations for high dimensional variable selection. What limits of the dimensionality such methods can handle, what the role of penalty functions is, and what the statistical properties are rapidly drive the advances of the field. The properties of non-concave penalized likelihood and its roles in high dimensional statistical modeling are emphasized. We also review some recent advances in ultra-high dimensional variable selection, with emphasis on independence screening and two-scale methods.