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We consider the problem of estimating an unknown signal $x_0$ from noisylinear observations $y = Ax_0 + z\in R^m$. In many practical instances, $x_0$has a certain structure that can be captured by a structure inducing convexfunction $f(\cdot)$. For example, $\ell_1$ norm can be used to encourage asparse solution. To estimate $x_0$ with the aid of $f(\cdot)$, we consider thewell-known LASSO method and provide sharp characterization of its performance.We assume the entries of the measurement matrix $A$ and the noise vector $z$have zero-mean normal distributions with variances $1$ and $\sigma^2$respectively. For the LASSO estimator $x^*$, we attempt to calculate theNormalized Square Error (NSE) defined as $\frac{\|x^*-x_0\|_2^2}{\sigma^2}$ asa function of the noise level $\sigma$, the number of observations $m$ and thestructure of the signal. We show that, the structure of the signal $x_0$ andchoice of the function $f(\cdot)$ enter the error formulae through the summaryparameters $D(cone)$ and $D(\lambda)$, which are defined as the Gaussiansquared-distances to the subdifferential cone and to the $\lambda$-scaledsubdifferential, respectively. The first LASSO estimator assumes a-prioriknowledge of $f(x_0)$ and is given by $\arg\min_{x}\{{\|y-Ax\|_2}~\text{subjectto}~f(x)\leq f(x_0)\}$. We prove that its worst case NSE is achieved when$\sigma\rightarrow 0$ and concentrates around $\frac{D(cone)}{m-D(cone)}$.Secondly, we consider $\arg\min_{x}\{\|y-Ax\|_2+\lambda f(x)\}$, for some$\lambda\geq 0$. This time the NSE formula depends on the choice of $\lambda$and is given by $\frac{D(\lambda)}{m-D(\lambda)}$. We then establish a mappingbetween this and the third estimator $\arg\min_{x}\{\frac{1}{2}\|y-Ax\|_2^2+\lambda f(x)\}$. Finally, for a number of important structured signal classes,we translate our abstract formulae to closed-form upper bounds on the NSE.

The Squared-Error of Generalized LASSO: A Precise Analysis