In the graph signal processing literature, most methods were developed based on the assumption of Gaussian noise since it can lead to computationally efficient and mathematically tractable solutions. Unfortunately, the Gaussian distribution cannot capture the sharp spikes and tail heaviness of the signal noise in various natural phenomena. The α-stable distribution is a generalization of the Gaussian distribution and is a more appropriate model for such impulsive heavy-tailed noise. In this paper, we consider the problem of adaptive estimation and sparse sampling for signals defined over graphs in the presence of impulsive α-stable noise. To tackle the problems associated with α-stable noise, the graph signal estimation problem is formulated as a minimum dispersion (MD)-based optimization. A novel adaptive least mean pth power (LMP) algorithm is proposed for robust estimation of band-limited graph signals from partial observations in α-stable noise environments. The mean square performance of the proposed LMP algorithm is theoretically analyzed. To handle the case of band-limited graph signals with unknown and time-varying bandwidth and spectral contents, the sparse nature of the Graph Fourier transform of the signal is exploited to develop an adaptive graph sampling technique. Specifically, the proposed sparse sampling technique identifies the spectral signal support via sparse online estimation building on the iterative shrinkage-thresholding algorithm. Numerical simulation studies are presented to corroborate the performance advantages of the proposed adaptive estimation and sparse sampling algorithms.

在图形信号处理文献中，大多数方法都是基于高斯噪声的假设而开发的，因为它会导致计算效率高且数学上易于处理的解决方案。不幸的是，高斯分布无法捕获各种自然现象中信号噪声的尖峰和尾部沉重。所述α -stable分布是高斯分布的概括，并且是这样的脉冲重尾噪声更合适的模型。在本文中，我们考虑在存在脉冲α稳定噪声的情况下，针对图上定义的信号进行自适应估计和稀疏采样的问题。解决与α相关的问题对于稳定的噪声，图形信号估计问题被表述为基于最小频散（MD）的优化。一种新的自适应最小均p次方（LMP）算法用于从在部分观测带限曲线信号的鲁棒估计α稳定的噪声环境。从理论上分析了提出的LMP算法的均方性能。为了处理具有未知且随时间变化的带宽和频谱内容的带限图形信号的情况，利用信号的图形傅里叶变换的稀疏性质来开发自适应图形采样技术。具体而言，所提出的稀疏采样技术通过基于迭代收缩阈值算法的稀疏在线估计来识别频谱信号支持。进行了数值模拟研究，以证实所提出的自适应估计和稀疏采样算法的性能优势。