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The Squared-Error of Generalized LASSO: A Precise Analysis

We consider the problem of estimating an unknown signal $x_0$ from noisy linear 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 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 $σ^2$ respectively. For the LASSO estimator $x^*$, we attempt to calculate the Normalized Square Error (NSE) defined as $\frac{\|x^*-x_0\|_2^2}{σ^2}$ as a function of the noise level $σ$, 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 $D(λ)$, which are defined as the Gaussian squared-distances to the subdifferential cone and to the $λ$-scaled subdifferential, respectively. The first LASSO estimator assumes a-priori knowledge of $f(x_0)$ and is given by $\arg\min_{x}\{{\|y-Ax\|_2}~\text{subject to}~f(x)\leq f(x_0)\}$. We prove that its worst case NSE is achieved when $σ\rightarrow 0$ and concentrates around $\frac{D(cone)}{m-D(cone)}$. Secondly, we consider $\arg\min_{x}\{\|y-Ax\|_2+λf(x)\}$, for some $λ\geq 0$. This time the NSE formula depends on the choice of $λ$ and is given by $\frac{D(λ)}{m-D(λ)}$. We then establish a mapping between this and the third estimator $\arg\min_{x}\{\frac{1}{2}\|y-Ax\|_2^2+ λf(x)\}$. Finally, for a number of important structured signal classes, we translate our abstract formulae to closed-form upper bounds on the NSE.