The Squared-Error of Generalized LASSO: A Precise Analysis

Abstract: We consider the problem of estimating an unknown signal $x_0$ from noisy linear observations $y = Ax_0 + z\in R<sup>m$. In many practical instances, $x_0$ has a certain structure that can be captured by a structure inducing convex function $f(\cdot)$. For example, $\ell_1$ norm can be used to encourage a sparse solution. To estimate $x_0$ with the aid of $f(\cdot)$, we consider the well-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 $\sigma<sup>2$ respectively. For the LASSO estimator $x<sup>*$, we attempt to calculate the Normalized Square Error (NSE) defined as $\frac{|x<sup>*-x_0|_2<sup>2}{\sigma<sup>2}$ as a function of the noise level $\sigma$, the number of observations $m$ and the structure of the signal. We show that, the structure of the signal $x_0$ and choice of the function $f(\cdot)$ enter the error formulae through the summary parameters $D(cone)$ and $D(\lambda)$, which are defined as the Gaussian squared-distances to the subdifferential cone and to the $\lambda$-scaled subdifferential, respectively. The first LASSO estimator assumes a-priori knowledge of $f(x_0)$ and is given by $$\arg\min_{x}{{|y-Ax|<em>2}~\text{subject to}~f(x)\leq f(x_0)}$$. 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 $\arg\min</em>{x}{|y-Ax|<em>2+\lambda f(x)}$, 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 mapping between this and the third estimator $$\arg\min</em>{x}{\frac{1}{2}|y-Ax|_2²⁺ \lambda f(x)}$$. Finally, for a number of important structured signal classes, we translate our abstract formulae to closed-form upper bounds on the NSE.