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Recent progress on variable projection methods for structured lowrank approximation
 Signal Processing
, 2014
"... Rank deficiency of a data matrix is equivalent to the existence of an exact linear model for the data. For the purpose of linear static modeling, the matrix is unstructured and the corresponding modeling problem is an approximation of the matrix by another matrix of a lower rank. In the context of l ..."
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Rank deficiency of a data matrix is equivalent to the existence of an exact linear model for the data. For the purpose of linear static modeling, the matrix is unstructured and the corresponding modeling problem is an approximation of the matrix by another matrix of a lower rank. In the context of linear timeinvariant dynamic models, the appropriate data matrix is Hankel and the corresponding modeling problems becomes structured lowrank approximation. Lowrank approximation has applications in: system identification; signal processing, machine learning, and computer algebra, where different types of structure and constraints occur. This paper gives an overview of recent progress in efficient local optimization algorithms for solving weighted mosaicHankel structured lowrank approximation problems. In addition, the data matrix may have missing elements and elements may be specified as exact. The described algorithms are implemented in a publicly available software package. Their application to system identification, approximate common divisor, and datadriven simulation problems is described in this paper and is illustrated by reproducible simulation examples. As a data modeling paradigm the lowrank approximation setting is closely related to the the behavioral approach in systems and control, total least squares, errorsinvariables modeling, principal component analysis, and rank minimization.
Chapter 1 Bounded Matrix Low Rank Approximation
"... It is common in recommender systems rating matrix, where the input matrix R is bounded in between [rmin, rmax] such as [1, 5]. In this chapter, we propose a new improved scalable low rank approximation algorithm for such bounded matrices called Bounded Matrix Low Rank Approximation(BMA) that bounds ..."
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It is common in recommender systems rating matrix, where the input matrix R is bounded in between [rmin, rmax] such as [1, 5]. In this chapter, we propose a new improved scalable low rank approximation algorithm for such bounded matrices called Bounded Matrix Low Rank Approximation(BMA) that bounds every element of the approximation PQ. We also present an alternate formulation to bound existing recommender system algorithms called BALS and discuss its convergence. Our experiments on real world datasets illustrate that the proposed method BMA outperforms the state of the art algorithms for recommender system such as Stochastic Gradient Descent, Alternating Least Squares with regularization, SVD++ and BiasSVD on real