Parametric Minimum Error Entropy Criterion: A Case Study in Blind Sensor Fusion and Regression Problems

The purpose of this article is to present the Parametric Minimum Error Entropy (PMEE) principle and to show a case study of the proposed criterion in a blind sensor fusion and regression problem. This case study consists on the estimation of a temporal series with a certain temporal invariance, which is measured from multiple independent sensors with unknown variances and unknown mutual correlations of the measurement errors. In this setting, we show that a particular case of the PMEE criterion is obtained from the Conditional Maximum Likelihood (CML) principle of the measurement model, leading to a semi-data-driven solution. Despite the fact that Information Theoretic Criteria (ITC) are inherently robust, they often result in difficult non-convex optimization problems. Our proposal is to address the non-convexity by means of a Majorization-Minimization (MM) based algorithm. We prove the conditions in which the resulting solution of the proposed algorithm reaches a stationary point of the original problem. In fact, the aforementioned global convergence of the proposed algorithm is possible thanks to a reformulation of the original cost function in terms of a variable constrained in the Grassmann manifold. As shown in this paper, the latter procedure is possible thanks to a homogeneity property of the PMEE cost function.