标题： 相干成像中估计问题的极小极大分析 显示英文标题

标题： Minimax Analysis of Estimation Problems in Coherent Imaging

摘要： 与磁共振成像等传统成像方式不同，这些方式通常可以用线性回归框架很好地描述，相干成像系统遵循一个显著更复杂的模型。 在这些系统中，任务是从形式为\[ {\boldsymbol y}_l = A_l X_o {\boldsymbol w}_l + {\boldsymbol z}_l, \quad l = 1, \ldots, L, \]的观测值${\boldsymbol y}_1, \ldots, {\boldsymbol y}_L \in \mathbb{R}^m$中估计未知图像${\boldsymbol x}_o \in \mathbb{R}^n$，其中$X_o = \mathrm{diag}({\boldsymbol x}_o)$是一个$n \times n$对角矩阵，${\boldsymbol w}_1, \ldots, {\boldsymbol w}_L \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,I_n)$表示散斑噪声，而${\boldsymbol z}_1, \ldots, {\boldsymbol z}_L \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,σ_z^2 I_m)$表示加性噪声。 矩阵$A_1, \ldots, A_L$是由成像系统确定的前向算子。 传统成像系统的极限通过稀疏线性回归模型得到了广泛研究。 然而，相干成像系统的极限仍大多未被探索。 我们的目标是通过表征高维设置中估计${\boldsymbol x}_o$的最小最大风险来填补这一空白。 受稀疏回归见解的启发，我们观察到${\boldsymbol x}_o$的结构在决定估计误差中起着关键作用。 在本工作中，我们采用基于覆盖数的一般结构概念，这更适合于相干成像系统。 我们证明最小最大均方误差（MSE）按\[ \frac{\max\{σ_z^4,\, m^2,\, n^2\}\, k \log n}{m^2 n L}, \]缩放，其中$k$是一个量化图像类有效复杂度的参数。 显示英文摘要

摘要： 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 ${\boldsymbol x}_o \in \mathbb{R}^n$ from observations ${\boldsymbol y}_1, \ldots, {\boldsymbol y}_L \in \mathbb{R}^m$ of the form \[ {\boldsymbol y}_l = A_l X_o {\boldsymbol w}_l + {\boldsymbol z}_l, \quad l = 1, \ldots, L, \] where $X_o = \mathrm{diag}({\boldsymbol x}_o)$ is an $n \times n$ diagonal matrix, ${\boldsymbol w}_1, \ldots, {\boldsymbol w}_L \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,I_n)$ represent speckle noise, and ${\boldsymbol z}_1, \ldots, {\boldsymbol z}_L \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,σ_z^2 I_m)$ denote additive noise. The matrices $A_1, \ldots, A_L$ 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 remain largely unexplored. Our goal is to close this gap by characterizing the minimax risk of estimating ${\boldsymbol x}_o$ in high-dimensional settings. Motivated by insights from sparse regression, we observe that the structure of ${\boldsymbol x}_o$ 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 \[ \frac{\max\{σ_z^4,\, m^2,\, n^2\}\, k \log n}{m^2 n L}, \] where $k$ is a parameter that quantifies the effective complexity of the class of images.