Constructing fixed rank optimal estimators with method of best recurrent approximations

We propose a new approach which generalizes and improves principal component analysis (PCA) and its recent advances. The approach is based on the following underlying ideas. PCA can be reformulated as a technique which provides the best linear estimator of the fixed rank for random vectors. By the proposed method, the vector estimate is presented in a special quadratic form aimed to improve the error of estimation compared with customary linear estimates. The vector is first pre-estimated from the special iterative procedure such that each iterative loop consists of a solution of the unconstrained nonlinear best approximation problem. Then, the final vector estimate is obtained from a solution of the constrained best approximation problem with the quadratic approximant. We show that the combination of these techniques allows us to provide a new nonlinear estimator with a significantly better performance compared with that of PCA and its known modifications.