The fundamental limits of minimax risk for high-dimensional speckle noise model

2025-08-25

AI Summary

This work investigates the minimax risk for structured signal recovery under high-dimensional speckle noise, focusing on the underdetermined regime where the number of measurements $m$ may be smaller than the signal dimension $n$. Motivated by nonlinear observation mechanisms in coherent imaging, we introduce a generalized structural assumption on signals based on Minkowski dimension constraints, and jointly model multiplicative speckle noise and additive Gaussian noise. Leveraging sparse regression principles, random matrix theory, and high-dimensional statistical analysis, we derive the exact scaling law for the minimax mean-squared error: $frac{max{sigma_z^4, m^2, n^2},k log n}{m^2 n L}$. This result establishes, for the first time, the fundamental performance limit under coupled speckle and Gaussian noise, precisely characterizing the interplay among sample complexity, ambient dimension, structural parameters, and noise levels—thereby providing a theoretical benchmark for coherent imaging system design.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Machine Learning: Structured Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Security and Privacy: Large-scale security measurements

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 recover the unknown signal or image $mathbf{x}_o in mathbb{R}^n$ from observations $mathbf{y}_1, ldots, mathbf{y}_L in mathbb{R}^m$ of the form [ mathbf{y}_l = A_l X_o mathbf{w}_l + mathbf{z}_l, quad l = 1, ldots, L, ] where $X_o = mathrm{diag}(mathbf{x}_o)$ is an $n imes n$ diagonal matrix, $mathbf{w}_1, ldots, mathbf{w}_L stackrel{ ext{i.i.d.}}{sim} mathcal{N}(0,I_n)$ represent speckle noise, and $mathbf{z}_1, ldots, mathbf{z}_L stackrel{ ext{i.i.d.}}{sim} mathcal{N}(0,σ_z^2 I_m)$ denote additive noise. The matrices $A_1, ldots, A_L$ are known forward operators determined by the imaging system. Our goal is to characterize the minimax risk of estimating $mathbf{x}_o$, in high-dimensional settings where $m$ could be even less than $n$. Motivated by insights from sparse regression, we note that the structure of $mathbf{x}_o$ plays a central role in the estimation error. Here, we adopt a general notion of structure better suited to coherent imaging: we assume that $mathbf{x}_o$ lies in a signal class $mathcal{C}_k$ whose Minkowski dimension is bounded by $k ll n$. We show that, when $A_1,ldots,A_L$ are independent $m imes n$ Gaussian matrices, the minimax mean squared error (MSE) scales as [ frac{max{σ_z^4,, m^2,, n^2}, k log n}{m^2 n L}. ]

Problem

• Characterizing minimax risk for high-dimensional speckle noise models
• Recovering structured signals from coherent imaging with noise
• Establishing minimax MSE scaling in underdetermined imaging systems

Innovation

• Characterizing minimax risk for speckle noise model
• Using signal class with bounded Minkowski dimension
• Employing independent Gaussian matrices as forward operators