  The class of Lq-regularized least squares (LQLS) are considered for
estimating a p-dimensional vector \b{eta} from its n noisy linear observations
y = X\b{eta}+w. The performance of these schemes are studied under the
high-dimensional asymptotic setting in which p grows linearly with n. In this
asymptotic setting, phase transition diagrams (PT) are often used for comparing
the performance of different estimators. Although phase transition analysis is
shown to provide useful information for compressed sensing, the fact that it
ignores the measurement noise not only limits its applicability in many
application areas, but also may lead to misunderstandings. For instance,
consider a linear regression problem in which n > p and the signal is not
exactly sparse. If the measurement noise is ignored in such systems,
regularization techniques, such as LQLS, seem to be irrelevant since even the
ordinary least squares (OLS) returns the exact solution. However, it is
well-known that if n is not much larger than p then the regularization
techniques improve the performance of OLS. In response to this limitation of PT
analysis, we consider the low-noise sensitivity analysis. We show that this
analysis framework (i) reveals the advantage of LQLS over OLS, (ii) captures
the difference between different LQLS estimators even when n > p, and (iii)
provides a fair comparison among different estimators in high signal-to-noise
ratios. As an application of this framework, we will show that under mild
conditions LASSO outperforms other LQLS even when the signal is dense. Finally,
by a simple transformation we connect our low-noise sensitivity framework to
the classical asymptotic regime in which n/p goes to infinity and characterize
how and when regularization techniques offer improvements over ordinary least
squares, and which regularizer gives the most improvement when the sample size
is large.
