2025/08/25 by Xing, Hao, Jana, Soham, Maleki, Arian
#FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2508.18503
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 xo ∈ ℝn from observations \boldsymbol y1, …, \boldsymbol yL ∈ ℝm of the form \boldsymbol yl = Al Xo \boldsymbol wl + \boldsymbol zl, l = 1, …, L, where Xo = diag(\boldsymbol xo) is an n × n diagonal matrix, \boldsymbol w1, …, \boldsymbol wL \stackreli.i.d.∼ N(0,In) represent speckle noise, and \boldsymbol z1, …, \boldsymbol zL \stackreli.i.d.∼ N(0,σz2 Im) denote additive noise. The matrices A1, …, AL 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 xo in high-dimensional settings. Motivated by insights from sparse regression, we observe that the structure of \boldsymbol xo 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 \fracmax\σz4, m2, n2\ k log nm2 n L, where k is a parameter that quantifies the effective complexity of the class of images.