# Convex Parameter Estimation of Perturbed Multivariate Generalized Gaussian Distributions

arXiv: https://arxiv.org/abs/2312.07479 · HTML: https://mlsanity.com/paper/2312.07479/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 50 references that are in the map.
- First posted on arXiv 12 Dec 2023.

## Builds on (most cited first, 10 of 10)

- [Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems](https://mlsanity.com/paper/1406.5429/) (2014)
- [Fixed Point Strategies in Data Science](https://mlsanity.com/paper/2008.02260/) (2020)
- [Regularized Tyler's Scatter Estimator: Existence, Uniqueness, and Algorithms](https://mlsanity.com/paper/1407.3079/) (2014)
- [Regularized M-estimators of scatter matrix](https://mlsanity.com/paper/1405.2528/) (2014)
- [A flexible EM-like clustering algorithm for noisy data](https://mlsanity.com/paper/1907.01660/) (2019)
- [Multivariate Generalized Gaussian Distribution: Convexity and Graphical Models](https://mlsanity.com/paper/1304.3206/) (2013)
- [Robust high-dimensional precision matrix estimation](https://mlsanity.com/paper/1501.01219/) (2015)
- [Robust estimation of precision matrices under cellwise contamination](https://mlsanity.com/paper/1501.01950/) (2015)
- [Robust classification with flexible discriminant analysis in heterogeneous data](https://mlsanity.com/paper/2201.02967/) (2022)
- [Perspective Functions: Proximal Calculus and Applications in High-Dimensional Statistics](https://mlsanity.com/paper/1610.01478/) (2016)

## Abstract (arXiv)

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.