Sharp MSE Bounds for Proximal Denoising

Abstract: Denoising has to do with estimating a signal $x_0$ from its noisy observations $y=x_0+z$. In this paper, we focus on the "structured denoising problem", where the signal $x_0$ possesses a certain structure and $z$ has independent normally distributed entries with mean zero and variance $\sigma<sup>2$. We employ a structure-inducing convex function $f(\cdot)$ and solve $$\min_x{\frac{1}{2}|y-x|<em>2^{2+\sigma\lambda} f(x)}$$ to estimate $x_0$, for some $\lambda&gt;0$. Common choices for $f(\cdot)$ include the $\ell_1$ norm for sparse vectors, the $\ell_1-\ell_2$ norm for block-sparse signals and the nuclear norm for low-rank matrices. The metric we use to evaluate the performance of an estimate $x<sup>*$ is the normalized mean-squared-error $$\text{NMSE}(\sigma)=\frac{\mathbb{E}|x<sup>*-x_0|_2<sup>2}{\sigma<sup>2}$$. We show that NMSE is maximized as $\sigma\rightarrow 0$ and we find the \emph{exact} worst case NMSE, which has a simple geometric interpretation: the mean-squared-distance of a standard normal vector to the $\lambda$-scaled subdifferential $\lambda\partial f(x_0)$. When $\lambda$ is optimally tuned to minimize the worst-case NMSE, our results can be related to the constrained denoising problem $\min</em>{f(x)\leq f(x_0)}{|y-x|<em>2}$. The paper also connects these results to the generalized LASSO problem, in which, one solves $\min</em>{f(x)\leq f(x_0)}{|y-Ax|_2}$ to estimate $x_0$ from noisy linear observations $y=Ax_0+z$. We show that certain properties of the LASSO problem are closely related to the denoising problem. In particular, we characterize the normalized LASSO cost and show that it exhibits a "phase transition" as a function of number of observations. Our results are significant in two ways. First, we find a simple formula for the performance of a general convex estimator. Secondly, we establish a connection between the denoising and linear inverse problems.