高维情形下尖峰混合模型协方差特征值的相变

中文摘要 AI 辅助

尖峰混合模型（SMM）作为一种概率模型，将单尖峰（威沙特）模型推广到混合模型形式。其应用广泛，从生命科学中的成像质谱到计算机视觉中的高光谱成像。理解在何种情况下能从噪声测量中恢复信号至关重要，且此类测量具有高复用性，需在高维环境下分析。本文证明了SMM协方差矩阵的极端特征值在高维情形下呈现相变。该相变及通过极端特征值进行的信号恢复取决于多个相互作用因素：尖峰间的相关性、能量参数和混合概率。此工作给出了从SMM协方差矩阵极端特征值检测一个或多个尖峰所需参数的精确信息论界限，这可能影响SMM的任何应用，理解这种相互作用可为分析化学和生命科学中的实验设计提供工具。

英文摘要

The spiked mixture model (SMM) has been introduced as a probabilistic model that generalizes the single-spike (Wishart) model to a mixture model form. With applications ranging from imaging mass spectrometry in the life sciences to hyperspectral imaging in computer vision, it is crucial to understand under which circumstances its signals can be recovered from noisy measurements. The highly multiplexed nature of these measurement types furthermore necessitates such analysis to hold in highdimensional settings. In this paper, we prove that the extreme eigenvalues of the covariance matrix from the SMM exhibit a phase transition in high-dimensional regimes. We show that this phase transition, and thus signal recovery by extreme eigenvalues, depends on several interacting factors: the correlation between spikes (i.e., how similar in content underlying signals are), the energy parameters (i.e., the absolute strength of each underlying signal), and the mixture probabilities (i.e., how likely it is to encounter each underlying signal). This work provides sharp information-theoretic bounds on the parameters needed to detect one or more spikes from extreme eigenvalues of the SMM covariance matrix, and these guarantees could potentially impact any application of the SMM. Understanding this interplay could serve as a tool for driving experimental design in analytical chemistry and life sciences.

发表机构

• Delft University of Technology（代尔夫特理工大学）
• Vanderbilt University（范德堡大学）
• Vanderbilt University School of Medicine（范德堡大学医学院）

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