  Given an unknown signal $\mathbf{x}_0\in\mathbb{R}^n$ and linear noisy
measurements
$\mathbf{y}=\mathbf{A}\mathbf{x}_0+\sigma\mathbf{v}\in\mathbb{R}^m$, the
generalized $\ell_2^2$-LASSO solves
$\hat{\mathbf{x}}:=\arg\min_{\mathbf{x}}\frac{1}{2}\|\mathbf{y}-\mathbf{A}\mathbf{x}\|_2^2
+ \sigma\lambda f(\mathbf{x})$. Here, $f$ is a convex regularization function
(e.g. $\ell_1$-norm, nuclear-norm) aiming to promote the structure of
$\mathbf{x}_0$ (e.g. sparse, low-rank), and, $\lambda\geq 0$ is the regularizer
parameter. A related optimization problem, though not as popular or well-known,
is often referred to as the generalized $\ell_2$-LASSO and takes the form
$\hat{\mathbf{x}}:=\arg\min_{\mathbf{x}}\|\mathbf{y}-\mathbf{A}\mathbf{x}\|_2 +
\lambda f(\mathbf{x})$, and has been analyzed in [1]. [1] further made
conjectures about the performance of the generalized $\ell_2^2$-LASSO. This
paper establishes these conjectures rigorously. We measure performance with the
normalized squared error
$\mathrm{NSE}(\sigma):=\|\hat{\mathbf{x}}-\mathbf{x}_0\|_2^2/\sigma^2$.
Assuming the entries of $\mathbf{A}$ and $\mathbf{v}$ be i.i.d. standard
normal, we precisely characterize the "asymptotic NSE"
$\mathrm{aNSE}:=\lim_{\sigma\rightarrow 0}\mathrm{NSE}(\sigma)$ when the
problem dimensions $m,n$ tend to infinity in a proportional manner. The role of
$\lambda,f$ and $\mathbf{x}_0$ is explicitly captured in the derived expression
via means of a single geometric quantity, the Gaussian distance to the
subdifferential. We conjecture that $\mathrm{aNSE} =
\sup_{\sigma>0}\mathrm{NSE}(\sigma)$. We include detailed discussions on the
interpretation of our result, make connections to relevant literature and
perform computational experiments that validate our theoretical findings.
