We discuss the inhomogeneous spiked Wigner model, a theoretical framework
recently introduced to study structured noise in various learning scenarios,
through the prism of random matrix theory, with a specific focus on its
spectral properties. Our primary objective is to find an optimal spectral
method and to extend the celebrated \cite{BBP} (BBP) phase transition criterion
-- well-known in the homogeneous case -- to our inhomogeneous,
block-structured, Wigner model. We provide a thorough rigorous analysis of a
transformed matrix and show that the transition for the appearance of 1) an
outlier outside the bulk of the limiting spectral distribution and 2) a
positive overlap between the associated eigenvector and the signal, occurs
precisely at the optimal threshold, making the proposed spectral method optimal
within the class of iterative methods for the inhomogeneous Wigner problem.

我们通过随机矩阵理论，重点研究了非齐次分组伪威格纳模型中的结构噪声，目标是找到一个最优的频谱方法，并将均匀情况下著名的 BBP 相变准则扩展到非齐次块状威格纳模型，我们对一个变换矩阵进行了彻底而严格的分析，并展示了在最佳阈值处 1）超出极限谱分布的异常值的出现，和 2）相应特征向量与信号之间的正重叠，在最佳阈值处同时出现，使得所提议的频谱方法在非齐次威格纳问题的迭代方法类中是最优的。

分块结构的特征相位转换与最优主成分分析

Spectral Phase Transition and Optimal PCA in Block-Structured Spiked  models

Quanta Image Sensor (QIS) is a binary imaging device envisioned to be the
next generation image sensor after CCD and CMOS. Equipped with a massive number
of single photon detectors, the sensor has a threshold $q$ above which the
number of arriving photons will trigger a binary response "1", or "0"
otherwise. Existing methods in the device literature typically assume that
$q=1$ uniformly. We argue that a spatially varying threshold can significantly
improve the signal-to-noise ratio of the reconstructed image. In this paper, we
present an optimal threshold design framework. We make two contributions.
First, we derive a set of oracle results to theoretically inform the maximally
achievable performance. We show that the oracle threshold should match exactly
with the underlying pixel intensity. Second, we show that around the oracle
threshold there exists a set of thresholds that give asymptotically unbiased
reconstructions. The asymptotic unbiasedness has a phase transition behavior
which allows us to develop a practical threshold update scheme using a
bisection method. Experimentally, the new threshold design method achieves
better rate of convergence than existing methods.

本文研究了基于单光子检测器的 Quanta 图像传感器，通过提出一种空间变化的阈值优化设计框架，实现对重建图像信噪比的提高，不仅从理论上证明了基于背景强度的最优阈值设计方案，还通过实验展示了该方法的收敛速度优于现有方法。