  We study the problem of estimating $\beta \in \mathbb{R}^p$ from its noisy
linear observations $y= X\beta+ w$, where $w \sim N(0, \sigma_w^2 I_{n\times
n})$, under the following high-dimensional asymptotic regime: given a fixed
number $\delta$, $p \rightarrow \infty$, while $n/p \rightarrow \delta$. We
consider the popular class of $\ell_q$-regularized least squares (LQLS)
estimators, a.k.a. bridge, given by the optimization problem: \begin{equation*}
\hat{\beta} (\lambda, q ) \in \arg\min_\beta \frac{1}{2} \|y-X\beta\|_2^2+
\lambda \|\beta\|_q^q, \end{equation*} and characterize the almost sure limit
of $\frac{1}{p} \|\hat{\beta} (\lambda, q )- \beta\|_2^2$. The expression we
derive for this limit does not have explicit forms and hence are not useful in
comparing different algorithms, or providing information in evaluating the
effect of $\delta$ or sparsity level of $\beta$. To simplify the expressions,
researchers have considered the ideal "no-noise" regime and have characterized
the values of $\delta$ for which the almost sure limit is zero. This is known
as the phase transition analysis.
  In this paper, we first perform the phase transition analysis of LQLS. Our
results reveal some of the limitations and misleading features of the phase
transition analysis. To overcome these limitations, we propose the study of
these algorithms under the low noise regime. Our new analysis framework not
only sheds light on the results of the phase transition analysis, but also
makes an accurate comparison of different regularizers possible.
