Robust Covariance Estimation for Entry-wise Contaminated and Partially Observed Data via Expectation Maximization Algorithm

Matys Savidan, Mohammed Nabil El Korso

Robust covariance estimation under entry-wise contamination and missing data is challenging in various signal processing applications. Under the Student-$t$ model, observation-level reweighting induces a uniform attenuation across samples, which can lead to cumulative information loss even when most coordinates remain informative. The alternative Student-$t^*$ model enables coordinate-wise robustness, but its E-step is not available in closed form. We extend inference for the Student-$t^*$ model to incomplete observations through a stochastic approximation EM (SAEM) algorithm operating on a reduced latent space: stochastic updates of the coordinate-wise scales are combined with Rao-Blackwellized Gaussian conditional moments, which avoids explicit sampling of latent Gaussian variables while preserving a likelihood-based framework. Experiments on synthetic and real sensor data show a lower Fisher-Rao error than Gaussian, Student-$t$, imputation-based and detect-then-estimate estimators at moderate to high outlier rates under diffuse contamination.