Compressible Distributions for High-dimensional Statistics

TL;DR

This paper investigates which probability distributions are compressible in high-dimensional settings, revealing that many popular sparse estimators like Lasso are ineffective for certain distributions such as Laplace, especially in large problem regimes.

Contribution

It introduces a principled framework to identify compressible distributions and demonstrates that common sparse regularization methods are often ineffective for distributions like Laplace in high dimensions.

Findings

Laplace distribution is incompressible in large GULR problems.

Least squares can outperform sparse estimators for certain distributions.

Rules of thumb based on moments characterize compressibility.

Abstract

We develop a principled way of identifying probability distributions whose independent and identically distributed (iid) realizations are compressible, i.e., can be well-approximated as sparse. We focus on Gaussian random underdetermined linear regression (GULR) problems, where compressibility is known to ensure the success of estimators exploiting sparse regularization. We prove that many distributions revolving around maximum a posteriori (MAP) interpretation of sparse regularized estimators are in fact incompressible, in the limit of large problem sizes. A highlight is the Laplace distribution and $\ell^{1}$ regularized estimators such as the Lasso and Basis Pursuit denoising. To establish this result, we identify non-trivial undersampling regions in GULR where the simple least squares solution almost surely outperforms an oracle sparse solution, when the data is generated from the Laplace distribution. We provide simple rules of thumb to characterize classes of compressible (respectively incompressible) distributions based on their second and fourth moments. Generalized Gaussians and generalized Pareto distributions serve as running examples for concreteness.