Konstantin Usevich

Structured low-rank approximation with missing data

We consider low-rank approximation of affinely structured matrices with missing elements. The method proposed is based on reformulation of the problem as inner and outer optimization. The inner minimization is a singular linear least-norm problem and admits an analytic solution. The outer problem is a nonlinear least-squares problem and is solved by local optimization methods: minimization subject to quadratic equality constraints and unconstrained minimization with regularized cost function. The method is generalized to weighted low-rank approximation with missing values and is illustrated on approximate low-rank matrix completion, system identification, and data-driven simulation problems. An extended version of this paper is a literate program, implementing the method and reproducing the presented results.

Multivariate and 2D Extensions of Singular Spectrum Analysis with the Rssa Package

Implementation of multivariate and 2D extensions of singular spectrum analysis (SSA) by means of the R package Rssa is considered. The extensions include MSSA for simultaneous analysis and forecasting of several time series and 2D-SSA for analysis of digital images. A new extension of 2D-SSA analysis called shaped 2D-SSA is introduced for analysis of images of arbitrary shape, not necessary rectangular. It is shown that implementation of shaped 2D-SSA can serve as a basis for implementation of MSSA and other generalizations. Efficient implementation of operations with Hankel and Hankel-block-Hankel matrices through the fast Fourier transform is suggested. Examples with code fragments in R, which explain the methodology and demonstrate the proper use of Rssa, are presented.

Structured Low-rank Approximation as a Rational Function Minimization

Abstract Many problems of system identification, model reduction and signal processing can be posed and solved as a structured low-rank approximation problem. In this paper a reformulation of the structured low-rank approximation problem as minimization of a multivariate rational cost function is considered. We show that in two different parametrizations the problem is reduced to optimization on a compact manifold or to a set of optimization problems on bounded domains of Euclidean space. We make a review of polynomial algebra methods for global optimization of the rational cost function.

Factorization Approach to Structured Low-Rank Approximation with Applications

We consider the problem of approximating an affinely structured matrix, for example a Hankel matrix, by a low-rank matrix with the same structure. This problem occurs in system identification, signal processing and computer algebra, among others. We impose the low-rank by modeling the approximation as a product of two factors with reduced dimension. The structure of the low-rank model is enforced by introducing a penalty term in the objective function. The proposed local optimization algorithm is able to solve the weighted structured low-rank approximation problem, as well as to deal with the cases of missing or fixed elements. In contrast to approaches based on kernel representations (in linear algebraic sense), the proposed algorithm is designed to address the case of small targeted rank. We compare it to existing approaches on numerical examples of system identification, approximate greatest common divisor problem, and symmetric tensor decomposition and demonstrate its consistently good performance.

A Recursive Restricted Total Least-Squares Algorithm

We show that the generalized total least squares (GTLS) problem with a singular noise covariance matrix is equivalent to the restricted total least squares (RTLS) problem and propose a recursive method for its numerical solution. The method is based on the generalized inverse iteration. The estimation error covariance matrix and the estimated augmented correction are also characterized and computed recursively. The algorithm is cheap to compute and is suitable for online implementation. Simulation results in least squares (LS), data least squares (DLS), total least squares (TLS), and restricted total least squares (RTLS) noise scenarios show fast convergence of the parameter estimates to their optimal values obtained by corresponding batch algorithms.

Variable projection methods for approximate (greatest) common divisor computations

We consider the problem of finding for a given

Adjusted least squares fitting of algebraic hypersurfaces

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Hankel Low-Rank Matrix Completion: Performance of the Nuclear Norm Relaxation

-tuple of polynomials (real or complex) the closest

Identifiability of an X-Rank Decomposition of Polynomial Maps

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Globally Convergent Jacobi-Type Algorithms for Simultaneous Orthogonal Symmetric Tensor Diagonalization

-tuple that has a common divisor of degree at least