On the Asymptotics of Maronna's Robust PCA

The eigenvalue decomposition (EVD) parameters of the second order statistics are ubiquitous in statistical analysis and signal processing. Notably, the EVD of the M-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 M-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 contributioninthe sense that it quantifies when itis acceptable to directly rely on well-established results on the EVD of Wishartdistributed matrix for characterizing the EVD of M-estimators. Finally, some examples (intrinsic bias analysis, rank estimation, and low-rank adaptive filtering) illustrate where the obtained results can be leveraged.