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Given an unknown signal $\mathbf{x}_0\in\mathbb{R}^n$ and linear noisymeasurements$\mathbf{y}=\mathbf{A}\mathbf{x}_0+\sigma\mathbf{v}\in\mathbb{R}^m$, thegeneralized $\ell_2^2$-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 regularizerparameter. 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 madeconjectures about the performance of the generalized $\ell_2^2$-LASSO. Thispaper establishes these conjectures rigorously. We measure performance with thenormalized 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. standardnormal, we precisely characterize the "asymptotic NSE"$\mathrm{aNSE}:=\lim_{\sigma\rightarrow 0}\mathrm{NSE}(\sigma)$ when theproblem 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 expressionvia means of a single geometric quantity, the Gaussian distance to thesubdifferential. We conjecture that $\mathrm{aNSE} =\sup_{\sigma>0}\mathrm{NSE}(\sigma)$. We include detailed discussions on theinterpretation of our result, make connections to relevant literature andperform computational experiments that validate our theoretical findings.

Asymptotically Exact Error Analysis for the Generalized $\ell_2^2$-LASSO