# Regularized M-estimators of scatter matrix

arXiv: https://arxiv.org/abs/1405.2528 · HTML: https://mlsanity.com/paper/1405.2528/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 May 2014.

## Built on by (most cited first, 11 of 11)

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- [Learning minimal volume uncertainty ellipsoids](https://mlsanity.com/paper/2405.02441/) (2024)
- [On a class of geodesically convex optimization problems solved via Euclidean MM methods](https://mlsanity.com/paper/2206.11426/) (2022)
- [Random matrix theory improved Fr'echet mean of symmetric positive definite matrices](https://mlsanity.com/paper/2405.06558/) (2024)
- [Disciplined Geodesically Convex Programming](https://mlsanity.com/paper/2407.05261/) (2024)
- [Structured Regularization for Constrained Optimization on the SPD Manifold](https://mlsanity.com/paper/2410.09660/) (2024)
- [Elliptical Wishart distributions: information geometry, maximum likelihood estimator, performance analysis and statistical learning](https://mlsanity.com/paper/2411.02726/) (2024)
- [Convex Parameter Estimation of Perturbed Multivariate Generalized Gaussian Distributions](https://mlsanity.com/paper/2312.07479/) (2023)
- [Efficient Estimation of Regularized Tyler's M-Estimator Using Approximate LOOCV](https://mlsanity.com/paper/2505.24781/) (2025)

## Abstract (arXiv)

In this paper, a general class of regularized M-estimators of scatter matrix are proposed which are suitable also for low or insufficient sample support (small n and large p) problems. The considered class constitutes a natural generalization of M-estimators of scatter matrix (Maronna, 1976) and are defined as a solution to a penalized M-estimation cost function that depend on a pair (α,β) of regularization parameters. We derive general conditions for uniqueness of the solution using concept of geodesic convexity. Since these conditions do not include Tyler's M-estimator, necessary and sufficient conditions for uniqueness of the penalized Tyler's cost function are established separately. For the regularized Tyler's M-estimator, we also derive a simple, closed form and data dependent solution for choosing the regularization parameter based on shape matrix matching in the mean squared sense. An iterative algorithm that converges to the solution of the regularized M-estimating equation is also provided. Finally, some simulations studies illustrate the improved accuracy of the proposed regularized M-estimators of scatter compared to their non-regularized counterparts in low sample support problems. An example of radar detection using normalized matched filter (NMF) illustrate that an adaptive NMF detector based on regularized M-estimators are able to maintain accurately the preset CFAR level and at at the same time provide similar probability of detection as the (theoretical) NMF detector.