Asymptotically exact error analysis for the generalized equation-LASSO
Paper i proceeding, 2015

Given an unknown signal x0 ϵ ℝn and linear noisy measurements y = Ax0 + σv ϵ ℝm, the generalized equation-LASSO solves equation. Here, f is a convex regularization function (e.g. ℓ1-norm, nuclear-norm) aiming to promote the structure of x0 (e.g. sparse, low-rank), and, λ ≥ 0 is the regularizer parameter. A related optimization problem, though not as popular or well-known, is often referred to as the generalized ℓ2-LASSO and takes the form equation, and has been analyzed by Oymak, Thrampoulidis and Hassibi. Oymak et al. further made conjectures about the performance of the generalized equation-LASSO. This paper establishes these conjectures rigorously. We measure performance with the normalized squared error equation. Assuming the entries of A are i.i.d. Gaussian N(0, 1/m) and those of v are i.i.d. N(0, 1), we precisely characterize the 'asymptotic NSE' aNSE :=limσ→0 NSE(σ) when the problem dimensions tend to infinity in a proportional manner. The role of λ, f and x0 is explicitly captured in the derived expression via means of a single geometric quantity, the Gaussian distance to the subdifferential. We conjecture that aNSE = supσ>0 NSE(σ). We include detailed discussions on the interpretation of our result, make connections to relevant literature and perform computational experiments that validate our theoretical findings.

Subdifferentials

Information theory

Optimization

Computational experiment

Noisy measurements

Optimization problems

Computation theory

Generalized Equations

Geometric quantities

Squared errors

Convex regularizations

Författare

C. Thrampoulidis

CalTech

Ashkan Panahi

Chalmers, Signaler och system, Signalbehandling och medicinsk teknik, Signalbehandling

B. Hassibi

CalTech

2015 IEEE International Symposium on Information Theory (ISIT)

2157-8095 (ISSN)

2021-2025

Ämneskategorier

Elektroteknik och elektronik

DOI

10.1109/ISIT.2015.7282810

ISBN

9781467377041