Best approximation of the identity mapping: The case of variable finite memory
Journal of Approximation Theory
Generalized Rank-Constrained Matrix Approximations
SIAM Journal on Matrix Analysis and Applications
Filtering and compression for infinite sets of stochastic signals
Signal Processing
Towards theory of generic Principal Component Analysis
Journal of Multivariate Analysis
Computational Methods for Modeling of Nonlinear Systems
Computational Methods for Modeling of Nonlinear Systems
IEEE Transactions on Signal Processing
IEEE Transactions on Signal Processing
Statistical analysis of subspace-based estimation of reduced-ranklinear regressions
IEEE Transactions on Signal Processing
Reduced-rank channel estimation for time-slotted mobile communication systems
IEEE Transactions on Signal Processing
The geometry of weighted low-rank approximations
IEEE Transactions on Signal Processing
Multislot estimation of fast-varying space-time communication channels
IEEE Transactions on Signal Processing
Reduced rank linear regression and weighted low rank approximations
IEEE Transactions on Signal Processing - Part I
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In this paper, a new theory of optimal weighted non-linear filtering is presented. Two filter models are considered. The first model is based on a representation of the filter in the polynomial-like form with q terms where each term consists of weighted matrices and the matrix determined from the error minimization problem. The second model extends the first one to the case of the filter concatenation. The filter models are given in terms of pseudo-inverse matrices, i.e., the requirement of invertibility for covariance matrices is omitted. Thus, our filters always exist. We develop methods which allow us to exploit advantages associated with the proposed nonlinear filter models. The methods consist of the orthogonalization procedure and the reduction of the original problem to q individual minimization problems for smaller matrices. This leads to a considerable reduction in the required computational work. The error associated with the first filter model decreases when the number q of terms of filter increases. Its compression ratio can be adjusted by varying a particular value of ranks in each of its q terms. This means that the proposed filer structure provides the two degrees of freedom. The second filter model provides another degree of freedom, a number k of filters in the concatenation. Variations of the degrees of freedom improve the performance of the proposed filters. In particular, the error associated with the filter concatenation decreases as the filter number k in the concatenation increases.