Joint Screening Tests for Lasso

Abstract

This paper focusses on "safe" screening techniques for the LASSO problem. Motivated by the need for low-complexity algorithms, we propose a new approach, dubbed "joint screening test", allowing to screen a set of atoms by carrying out one single test. The approach is particularized to two different sets of atoms, respectively expressed as sphere and dome regions. After presenting the mathematical derivations of the tests, we elaborate on their relative effectiveness and discuss the practical use of such procedures.

Gather and Conquer: Region-Based Strategies to Accelerate Safe Screening Tests

“…Another question of interest is the relative effectiveness of the joint sphere and dome tests proposed in (22) and (23)- (24), respectively. The next lemma provides some insights into this question.…”

“…Let us first consider the case when R l is a sphere, that is R l = B(t l , l ). We assume that the test vector t l is given and want to tune the value of l so that R l is the largest sphere passing (if possible) the joint screening test (22). Noticing that the joint sphere test (22) is satisfied as soon as the radius l verifies…”

“…Letting the radius l tend to its largest value t,c , we have that any atom a i ∈ A having a distance to t l strictly smaller than t l ,c will be screened by test (22), that is…”

“…This work is an extended version of [22] in which we laid the main foundations of joint screening. The present paper contains all the proofs of our results, extended discussions and developments, as well as extensive simulation results.…”

Atom Selection in Continuous Dictionaries: Reconciling Polar and SVD Approximations

“…Proof : Using (16), it is sufficient to show that the first three eigenfunctions of R (defined in (14)) can be written as linear combinations of cos(ωθ), sin(ωθ) and the constant function over Θ , say 1 Θ (θ). Now, using a simple trigonometric identity, we obtain κ(θ, θ ) = a + b cos(ωθ) cos(ωθ ) + b sin(ωθ) sin(ωθ ), and therefore span(R) = span(1 Θ (θ), cos(ωθ), sin(ωθ)).…”