[Paper Review] On the eigenvalues of the spatial sign covariance matrix in more than two dimensions

Alexander Dürre, David E. Tyler+1 more|arXiv (Cornell University)|Dec 9, 2015

Morphological variations and asymmetry22 references21 citations

TL;DR

This paper establishes that the eigenvalues of the spatial sign covariance matrix (SSCM) for elliptical distributions are a one-to-one function of the eigenvalues of the shape matrix, and that they are more tightly clustered than the original eigenvalues. It derives a one-dimensional integral representation for the SSCM eigenvalues, enabling efficient numerical computation and proving injectivity, ordering preservation, and eigenvalue clustering under elliptical symmetry in more than two dimensions.

ABSTRACT

We gather several results on the eigenvalues of the spatial sign covariance matrix of an elliptical distribution. It is shown that the eigenvalues are a one-to-one function of the eigenvalues of the shape matrix and that they are closer together than the latter. We further provide a one-dimensional integral representation of the eigenvalues, which facilitates their numerical computation.

Motivation & Objective

• To investigate the relationship between the eigenvalues of the spatial sign covariance matrix (SSCM) and the shape matrix in high-dimensional elliptical distributions.
• To establish that the SSCM eigenvalues are a one-to-one function of the shape matrix eigenvalues, ensuring identifiability.
• To demonstrate that SSCM eigenvalues are more tightly clustered (closer together) than the shape matrix eigenvalues.
• To derive a one-dimensional integral representation for SSCM eigenvalues to enable efficient numerical computation.
• To provide a theoretical foundation for robust multivariate analysis using spatial sign methods in high dimensions.

Proposed method

• The authors use the eigenvalue decomposition of the trace-normalized shape matrix $ V_0 = O\Lambda O^\top $, where $ \Lambda = \text{diag}(\lambda_1, \dots, \lambda_p) $ with $ \lambda_1 \geq \cdots \geq \lambda_p \geq 0 $ and $ \sum \lambda_j = 1 $.
• They express the population SSCM as $ S = O\Delta O^\top $, with $ \Delta = \text{diag}(\delta_1, \dots, \delta_p) $, where $ \delta_i = \mathbb{E}\left[ \lambda_i Y_i^2 \left( \sum_{j=1}^p \lambda_j Y_j^2 \right)^{-1} \right] $, and $ Y = RU $ has a spherical distribution.
• The authors exploit spherical symmetry of $ Y $ to choose a convenient distribution (uniform on the unit ball) for analytical tractability, simplifying the expectation integrals.
• They apply Gradshteyn and Ryzhik's formula 4.646 to derive one-dimensional integral representations for the eigenvalues $ \delta_i $, $ \eta_{ii} $, and $ \eta_{ij} $, enabling numerical evaluation.
• Injectivity of the mapping from $ \lambda $ to $ \delta $ is proven by contradiction, showing that different $ \lambda $ vectors yield different $ \delta $ vectors under the spherical symmetry assumption.
• The proof of eigenvalue clustering relies on showing that $ \delta_i $ values are strictly more equal than $ \lambda_i $, using inequalities based on weighted harmonic means and the strict inequality $ \rho_{(1)} < \rho_{(p)} $.

Experimental results

Research questions

• RQ1 Is the mapping from the shape matrix eigenvalues to the SSCM eigenvalues injective in dimensions greater than two?
• RQ2 How do the eigenvalues of the SSCM compare in spread to those of the shape matrix?
• RQ3 Can the eigenvalues of the SSCM be represented via a one-dimensional integral for efficient numerical computation?
• RQ4 What is the functional relationship between the eigenvalues of the SSCM and the shape matrix under elliptical symmetry?
• RQ5 Does the SSCM eigenvalue vector preserve the ordering of the shape matrix eigenvalues?

Key findings

• The mapping from shape matrix eigenvalues $ \lambda $ to SSCM eigenvalues $ \delta $ is injective, meaning $ \phi(\lambda) = \phi(\tilde{\lambda}) $ implies $ \lambda = \tilde{\lambda} $.
• The SSCM eigenvalues $ \delta_1, \dots, \delta_p $ are more tightly clustered than the shape matrix eigenvalues $ \lambda_1, \dots, \lambda_p $, i.e., $ \delta_1 - \delta_p < \lambda_1 - \lambda_p $.
• The SSCM eigenvalues $ \delta_i $ are a one-to-one function of the shape matrix eigenvalues and preserve their ordering: $ \delta_1 \geq \cdots \geq \delta_p $.
• A one-dimensional integral representation for $ \delta_i $ is derived using the uniform distribution on the unit ball and Gradshteyn and Ryzhik's formula 4.646, enabling numerical computation.
• The asymptotic covariance matrix of the SSCM estimator is expressed in terms of $ \delta_i $ and $ \eta_{ij} $, with $ \eta_{ij} $ also given by a one-dimensional integral.
• The eigenvalues $ \delta_i $ are expressed as $ \delta_i = \frac{p\Gamma(p/2)}{2\pi^{p/2}} \int_{S_{1,p}} \frac{\lambda_i z_i}{\sum_{j=1}^p \lambda_j z_j} \prod_{j=1}^p z_j^{-1/2} dz $, with $ S_{1,p} = \{ z \in \mathbb{R}^p_+ \mid \sum z_j \leq 1 \} $.