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.

中文翻译：

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扰动多元广义高斯分布的凸参数估计


多元广义高斯分布（MGGD），也称为多元指数幂（MEP）分布，广泛应用于信号和图像处理中。然而，估计实际应用中所需的 MGGD 参数仍然面临着特定的理论挑战。特别是，当分布均值和散布（或精度）矩阵都未知时，建立标准定点方法的收敛特性仍然是一个悬而未决的问题。在鲁棒估计中，对精度矩阵施加经典约束（例如稀疏性）受到所得成本函数的非凸性的限制。本文通过提出具有完善的收敛特性的凸公式，从优化的角度解决了这些问题。我们将分析嵌入到噪声场景中，其中通过对乘性扰动建模来引入鲁棒性。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。本文提出了所需理论属性的证明，指定了针对不同正则化选择保留这些属性的条件，并设计了通用的近端原始对偶优化策略。实验表明，与 Tyler 的 $M$ 估计器相比，平均向量参数的精度和协方差矩阵估计更加准确，且性能相似。在高维设置中，所提出的方法优于经典的 GLASSO（其鲁棒扩展之一）和正则化泰勒估计器。