Linear minimum mean-square error estimation based on high-dimensional data with missing values

Abstract

In linear minimum mean-square error (LMMSE) estimation problems, the observation data may have missing entries. Processing such data vectors may have high complexity if the observation data vector has high-dimensionality and the LMMSE estimator must be re-derived whenever there are missing values. In this context, a means of reducing the computational complexity is introduced when the number of missing entries is relatively small. All first- and second-order data statistics are assumed known, and the positions of the missing values are also known. The proposed method works by first applying the LMMSE estimator on the data vector with missing values replaced by zeros, and then applying a low-complexity update that depends on the positions of the missing. The method achieves exact LMMSE based on only observed data with lower complexity compared to the direct implementation of a time-varying LMMSE filter based on the incomplete data. We also show that if LMMSE imputation is used to fill the missing entires first based on the non-missing entries, and then a complete-data LMMSE filter is applied to the completed data vector, then the same linear MMSE is also achieved, but with higher complexity.