Convex Parameter Estimation of Perturbed Multivariate Generalized Gaussian Distributions

IF 4.6 2区 工程技术 Q1 ENGINEERING, ELECTRICAL & ELECTRONIC IEEE Transactions on Signal Processing Pub Date : 2024-09-04 DOI:10.1109/TSP.2024.3453509
Nora Ouzir;Frédéric Pascal;Jean-Christophe Pesquet
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Abstract

The multivariate generalized Gaussian distribution (MGGD), also known as the multivariate exponential power (MEP) distribution, is widely used in signal and image processing. However, estimating MGGD parameters, which is required in practical applications, still faces specific theoretical challenges. In particular, establishing convergence properties for the standard fixed-point approach when both the distribution mean and the scatter (or the precision) matrix are unknown is still an open problem. In robust estimation, imposing classical constraints on the precision matrix, such as sparsity, has been limited by the non-convexity of the resulting cost function. This paper tackles these issues from an optimization viewpoint by proposing a convex formulation with well-established convergence properties. We embed our analysis in a noisy scenario where robustness is induced by modelling multiplicative perturbations. The resulting framework is flexible as it combines a variety of regularizations for the precision matrix, the mean and model perturbations. This paper presents proof of the desired theoretical properties, specifies the conditions preserving these properties for different regularization choices and designs a general proximal primal-dual optimization strategy. The experiments show a more accurate precision and covariance matrix estimation with similar performance for the mean vector parameter compared to Tyler's $M$ -estimator. In a high-dimensional setting, the proposed method outperforms the classical GLASSO, one of its robust extensions, and the regularized Tyler's estimator.
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受扰动多变量广义高斯分布的凸参数估计
多元广义高斯分布(MGGD)又称多元指数幂(MEP)分布,在信号和图像处理中得到广泛应用。然而,实际应用中所需的 MGGD 参数估计仍面临特定的理论挑战。特别是,在分布均值和散点(或精度)矩阵都未知的情况下,建立标准定点方法的收敛特性仍是一个悬而未决的问题。在稳健估算中,对精度矩阵施加经典约束(如稀疏性)一直受限于由此产生的成本函数的非凸性。本文从优化的角度出发,提出了一种具有公认收敛特性的凸公式来解决这些问题。我们将分析嵌入到噪声场景中,通过模拟乘法扰动来实现鲁棒性。由此产生的框架非常灵活,因为它结合了精度矩阵、均值和模型扰动的各种正则化。本文证明了所需的理论特性,明确了在不同正则化选择下保持这些特性的条件,并设计了一种通用的近似原始二元优化策略。实验表明,与 Tyler 的 $M$-estimator 相比,平均向量参数的精度和协方差矩阵估计更为精确,性能相似。在高维环境下,所提出的方法优于经典的 GLASSO、其稳健扩展之一以及正则化泰勒估计器。
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来源期刊
IEEE Transactions on Signal Processing
IEEE Transactions on Signal Processing 工程技术-工程:电子与电气
CiteScore
11.20
自引率
9.30%
发文量
310
审稿时长
3.0 months
期刊介绍: The IEEE Transactions on Signal Processing covers novel theory, algorithms, performance analyses and applications of techniques for the processing, understanding, learning, retrieval, mining, and extraction of information from signals. The term “signal” includes, among others, audio, video, speech, image, communication, geophysical, sonar, radar, medical and musical signals. Examples of topics of interest include, but are not limited to, information processing and the theory and application of filtering, coding, transmitting, estimating, detecting, analyzing, recognizing, synthesizing, recording, and reproducing signals.
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