Minimax Analysis of Estimation Problems in Coherent Imaging

Hao Xing, Soham Jana, Arian Maleki

Abstract:

Unlike conventional imaging modalities, such as magnetic resonance imaging, which are often
well described by a linear regression framework, coherent imaging systems follow a significantly
more complex model. In these systems, the task is to estimate the unknown image $x_o\in R^n$
from observations $y_1,...,y_L \in R^m$ of the form
$y_l = A_l X_o w_l + z_l, l= 1,...,L,$
where $Xo = diag(x_o)$ is an n×n diagonal matrix, $w_1,...,w_L \sim N(0,I_n)$ represent i.i.d.speckle
noise, and $z_1,...,z_L \sim N(0,\sigma^2)$ denote i.i.d. additive noise. The matrices $A_1,...,A_L$ are known
forward operators determined by the imaging system.
The fundamental limits of conventional imaging systems have been extensively studied
through sparse linear regression models. However, the limits of coherent imaging systems re-
main largely unexplored. Our goal is to close this gap by characterizing the minimax risk of
estimating $x_o$ in high-dimensional settings.
Motivated by insights from sparse regression, we observe that the structure of $x_o$ plays
a crucial role in determining the estimation error. In this work, we adopt a general notion
of structure based on the covering numbers, which is more appropriate for coherent imaging
systems. We show that the minimax mean squared error (MSE) scales as
$\max{\sigma^4
z , m^2, n^2}k\log n\over
m^2nL $,
where $k$ is a parameter that quantifies the effective complexity of the class of images.