On robustness of Spectral Rényi divergence

Abstract: This paper studies a specific category of statistical divergences for spectral densities of time series: the spectral $\alpha$-R\'{e}nyi divergences, which includes the Itakura--Saito divergence as a subset. The aim of this paper is to highlight both information-theoretic and statistical properties of spectral $\alpha$-R\'{e}nyi divergences. We reveal the connection between the spectral $\alpha$-R\'{e}nyi divergence and the $\gamma$-divergence in robust statistics, and a variational representation of spectral $\alpha$-R\'{e}nyi divergence. Inspired by these results suggesting ``robustness'' of spectral $\alpha$-R\'{e}nyi divergence, we show that the minimum spectral R\'{e}nyi divergence estimate has a stable optimization path with respect to outliers in the frequency domain, unlike the minimum Itakura-Saito divergence estimator, and thus it delivers more stable estimate, reducing the need for intricate pre-processing.