A statistical mechanics approach to de-biasing and uncertainty estimation in LASSO for random measurements

TL;DR

This paper introduces a statistical mechanics-based method to de-bias LASSO estimators and estimate uncertainty in high-dimensional linear models with random measurement matrices, enabling hypothesis testing.

Contribution

It develops a computationally feasible scheme for de-biasing LASSO and constructing confidence intervals in high-dimensional settings with rotationally invariant measurement matrices.

Findings

Successfully de-biases LASSO estimators in experiments

Constructs confidence intervals and p-values for parameters

Validates approach with noisy measurement data

Abstract

In high-dimensional statistical inference in which the number of parameters to be estimated is larger than that of the holding data, regularized linear estimation techniques are widely used. These techniques have, however, some drawbacks. First, estimators are biased in the sense that their absolute values are shrunk toward zero because of the regularization effect. Second, their statistical properties are difficult to characterize as they are given as numerical solutions to certain optimization problems. In this manuscript, we tackle such problems concerning LASSO, which is a widely used method for sparse linear estimation, when the measurement matrix is regarded as a sample from a rotationally invariant ensemble. We develop a new computationally feasible scheme to construct a de-biased estimator with a confidence interval and conduct hypothesis testing for the null hypothesis that a certain parameter vanishes. It is numerically confirmed that the proposed method successfully de-biases the LASSO estimator and constructs confidence intervals and p-values by experiments for noisy linear measurements.