异方差噪声下的尖峰估计：基于随机分裂方法

中文摘要 AI 辅助

本文研究一类具有异方差且未知方差轮廓的尖峰Wigner型矩阵。众所周知，在BBP相变的超临界区域，强尖峰会在谱中产生离群值。然而，在异方差情形下，通常无法根据这些观测到的离群值一致地估计尖峰强度，因为后者是方差轮廓中未知参数的Dyson方程的解。受稀疏矩阵补全工作\citep{BordenaveCosteNadakuditi2023}的启发，本文通过将尖峰矩阵随机分裂为两部分引入一个非对称化模型，该模型将带噪的Wigner型矩阵转化为非Hermitian随机矩阵，同时以稀释为代价保留Hermitian尖峰。我们建立了该非对称化模型的BBP型相变，由此即使不知道噪声部分的方差轮廓，也能精确估计尖峰强度。我们进一步将方法应用于研究两个相关尖峰模型之间的相关性，其中两个模型的尖峰/信号部分相关，而噪声部分独立但可能均为异方差。通过分别及联合地对两个模型应用非对称化方法，我们能够获得两个模型信号部分之间相关性的精确估计。

英文摘要

In this paper, we consider a spiked Wigner type matrix with a heteroscedastic and unknown variance profile. It is well known that in the supercritical regime of the BBP transition, strong spikes can create outliers in the spectrum. Unfortunately, in the heteroscedastic case, in general it is not possible to estimate the spike strength from these observed outlier consistently, as the latter is a solution to a Dyson equation with unknown parameters from the variance profile. In this paper, inspired by the work on sparse matrix completion \citep{BordenaveCosteNadakuditi2023}, we introduce an asymmetrized model by randomly splitting the spiked matrix into two parts, which transforms the noisy Wigner type matrix into a non Hermitian random matrix, while preserving the Hermitian spikes at the cost of a dilution. We establish a BBP type transition for the asymmetrized model, from which we can estimate the strength of the spikes precisely, even without knowing the variance profile of the noise part. We then further apply our approach to study the correlation between two correlated spiked models, where the spike/signal parts of the two models are correlated, and the noise parts are independent but may both be heteroscedastic. By applying our asymmetrization approach to the two models separately and also jointly, we are able to obtain a precise estimate of the correlation between the signal parts of the two models.

发表机构

• University of Hong Kong（香港大学）
• Hong Kong University of Science and Technology（香港科技大学）

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