On the Asymptotics of Maronna's Robust PCA

• Published in: IEEE transactions on signal processing Vol. 67; no. 19; pp. 4964 - 4975
• Main Authors: Draskovic, Gordana, Breloy, Arnaud, Pascal, Frederic
• Format: Journal Article
• Language: English
• Published: IEEE 01.10.2019; Institute of Electrical and Electronics Engineers
• Subjects: <inline-formula xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <tex-math notation="LaTeX"> M</tex-math> </inline-formula>-estimators; Applications; CES distributions; Computer Science; Covariance matrices; eigenvalue decomposition; Eigenvalues and eigenfunctions; Estimation; Matrix decomposition; Other Statistics; Principal component analysis; principal subspace; Signal and Image Processing; Statistics; Symmetric matrices; Wishart distribution
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2019.2932877

The eigenvalue decomposition (EVD) parameters of the second order statistics are ubiquitous in statistical analysis and signal processing. Notably, the EVD of the <inline-formula><tex-math notation="LaTeX">M</tex-math></inline-formula>-estimators of the scatter matrix is a popular choice to perform robust probabilistic PCA or other dimension reduction related applications. Towards the goal of characterizing this process, this paper derives new asymptotics for the EVD parameters (i.e. eigenvalues, eigenvectors, and principal subspace) of <inline-formula><tex-math notation="LaTeX">M</tex-math></inline-formula>-estimators in the context of complex elliptically symmetric distributions. First, their Gaussian asymptotic distribution is obtained by extending standard results on the sample covariance matrix in the Gaussian context. Second, their convergence towards the EVD parameters of a Gaussian-Core Wishart Equivalent is derived. This second result represents the main contribution in the sense that it quantifies when it is acceptable to directly rely on well-established results on the EVD of Wishart-distributed matrix for characterizing the EVD of <inline-formula><tex-math notation="LaTeX">M</tex-math></inline-formula>-estimators. Finally, some examples (intrinsic bias analysis, rank estimation, and low-rank adaptive filtering) illustrate where the obtained results can be leveraged.