Mathematics > Statistics Theory
[Submitted on 7 Feb 2011 (v1), last revised 25 Apr 2012 (this version, v3)]
Title:Compressible Distributions for High-dimensional Statistics
View PDFAbstract: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.
Submission history
From: Remi Gribonval [view email] [via CCSD proxy][v1] Mon, 7 Feb 2011 09:14:13 UTC (217 KB)
[v2] Fri, 17 Jun 2011 20:19:50 UTC (250 KB)
[v3] Wed, 25 Apr 2012 12:53:23 UTC (792 KB)
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