Structured Low-Rank Approximation with Missing Data

Journal of Computational and Applied Mathematics

Usevich

2014

Self Cite

A software package is presented that computes locally optimal solutions to low-rank approximation problems with the following features:• mosaic Hankel structure constraint on the approximating matrix,• weighted 2-norm approximation criterion,• fixed elements in the approximating matrix,• missing elements in the data matrix, and• linear constraints on an approximating matrix's left kernel basis.It implements a variable projection type algorithm and allows the user to choose standard local optimization methods for the solution of the parameter optimization problem. For an m× n data matrix, with n > m, the computational complexity of the cost function and derivative evaluation is O(m 2 n). The package is suitable for applications with n ≫ m. In statistical estimation and data modeling-the main application areas of the package-n ≫ m corresponds to modeling of large amount of data by a low-complexity model. Performance results on benchmark system identification problems from the database DAISY and approximate common divisor problems are presented.

“…Fast methods for evaluation of f and its derivatives are presented in [39] and implemented in C++. The general case is a generalized least norm problem and is solved in [23].…”

Section: Solution Methodsmentioning

confidence: 99%

“…The software is written in C++ with Matlab/Octave and R interfaces and contains also the experimental Matlab code, presented in [23]. The latter supports general linear structure and missing values, but is inefficient and can be used only for small size matrices (say, m ≤ n < 100).…”

Section: Interfaces To Scientific Computing Environmentsmentioning

confidence: 99%

Software for weighted structured low-rank approximation

Journal of Computational and Applied Mathematics

Usevich

2014

Self Cite

“…In [11], it is sown how this problem can be solved in the presence of exact (v i = +∞) and missing (v i = 0) values. In [12], it is shown that fast O(T ) evaluation of M and its derivatives can be performed for mosaic-Hankel-like structured matrices.…”

Section: Solution Methodsmentioning

confidence: 99%

Approximate system identification with missing data

52nd IEEE Conference on Decision and Control

2013

Self Cite

Linear time-invariant system identification is considered in the behavioral setting. Nonstandard features of the problem are specification of missing and exact variables and identification from multiple time series with different length. The problem is equivalent to mosaic Hankel structured lowrank approximation with element-wise weighted cost function. Zero/infinite weights are assigned to the missing/exact data points. The problem is in general nonconvex. A solution method based on local optimization is outlined and compared with alternative methods on simulation examples.In a stochastic setting, the problem corresponds to errors-invariables identification. A modification of the generic problem considered is presented that is a deterministic equivalent to the classical ARMAX identification. The modification is also a mosaic Hankel structured low-rank approximation problem.

“…In total, 215 samples are missing in a periodic pattern, see Figure 1. The precise simulation parameters are specified in the following fragment of the m-file reproducing the presented numerical results: example +≡ (1,6), [], -1); T = 500; n = 6; w0 = initial(ss(sys0), ones(n, 1), T -1); Tm = sort(unique([1:7:T, 3:7:T, 5:7:T])); w = w0; w(Tm) = NaN; Two identification methods, described in the paper, are applied on the data and the results are validated by the distance between the characteristic polynomials of the true and identified models: example +≡ dist = @(sys1, sys2) norm(poly(eig(sys1))...…”

Section: Motivating Examplementioning

confidence: 99%

“…• methods based on convex relaxations [3], and • methods based on local optimization [4], [5], [6], [7]. To the best of our knowledge, the class of the subspace methods [8] has not been extended to deal with missing data.…”

Section: Introductionmentioning

confidence: 99%

Exact system identification with missing data

52nd IEEE Conference on Decision and Control

2013

Self Cite

The paper presents initial results on a subspace method for exact identification of a linear time-invariant system from data with missing values. The identification problem with missing data is equivalent to a Hankel structured lowrank matrix completion problem. The novel idea is to search systematically and use effectively completely specified submatrices of the incomplete Hankel matrix constructed from the given data. Nontrivial kernels of the rank-deficient completely specified submatrices carry information about the tobe-identified system. Combining this information into a full model of the identified system is a greatest common divisor computation problem. The developed subspace method has linear computational complexity in the number of data points and is therefore an attractive alternative to more expensive methods based on the nuclear norm heuristic.