Anatoli Torokhti

Generalized Rank-Constrained Matrix Approximations

In this paper we give an explicit solution to the rank-constrained matrix approximation in Frobenius norm, which is a generalization of the classical approximation of an

Towards theory of generic Principal Component Analysis

m\times n

An optimal filter of the second order

matrix

Optimal fixed rank transform of the second degree

A

Optimal transform formed by a combination of nonlinear operators: the case of data dimensionality reduction

by a matrix of, at most, rank

Weak interpolation and approximation of non-linear operators on the space C([0,1])

k

A methodology for the constructive approximation of nonlinear operators defined on noncompact sets

.

Method of recurrent best estimators of second degree for optimal filtering of random signals

In this paper, we consider a technique called the generic Principal Component Analysis (PCA) which is based on an extension and rigorous justification of the standard PCA. The generic PCA is treated as the best weighted linear estimator of a given rank under the condition that the associated covariance matrix is singular. As a result, the generic PCA is constructed in terms of the pseudo-inverse matrices that imply a development of the special technique. In particular, we give a solution of the new low-rank matrix approximation problem that provides a basis for the generic PCA. Theoretical aspects of the generic PCA are carefully studied.

Filtering and compression for infinite sets of stochastic signals

We present a new technique allowing us to find an optimal filter in the class of so-called second-order filters. The new filter is generated by a best-approximation operator of the second degree and generalizes and improves an optimal linear filter associated with the concept of Wiener filtering. This article provides a strict justification of the technique proposed, demonstrates its advantages, and describes numerous useful extensions and applications.

On Nonlinear Operator Approximation with Preassigned Accuracy

We present a new technique with potential applications to numerous areas in signal processing including data compression, filtering, blind channel equalization, and feature selection and classification in pattern recognition. The technique is based on the best constrained approximation of a stochastic signal by a specific second degree operator acting on the noisy observed data and therefore is called the second degree transform. The proposed transform generalizes an approach by Hua and Liu (1998) and gives a better performance in comparison. This paper provides a strict justification of the technique proposed and shows its useful application to data compression and filtering.