TY - JOUR
T1 - Interest zone matrix approximation
AU - Shabat, Gil
AU - Averbuch, Amir
N1 - Publisher Copyright:
© 2012, International Linear Algebra Society. All rights reserved.
PY - 2012
Y1 - 2012
N2 - An algorithm for matrix approximation, when only some of its entries are taken into consideration, is described. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank approximations, this type of algorithms appears recently in the literature under different names, where it usually uses the Expectation-Maximization algorithm that maximizes the likelihood for the missing entries. In this paper, the algorithm is extended to different cases other than low rank approximations under Frobenius norm, such as minimizing the Frobenius norm under nuclear norm constraint, spectral norm constraint, orthogonality constraint and more. The geometric interpretation of the proposed approximation process along with its optimality for convex constraints is also discussed. In addition, it is shown how the approximation algorithm can be used for matrix completion as well, under a variety of spectral regularizations. Its applications to physics, electrical engineering and data interpolation problems are also described.
AB - An algorithm for matrix approximation, when only some of its entries are taken into consideration, is described. The approximation constraint can be any whose approximated solution is known for the full matrix. For low rank approximations, this type of algorithms appears recently in the literature under different names, where it usually uses the Expectation-Maximization algorithm that maximizes the likelihood for the missing entries. In this paper, the algorithm is extended to different cases other than low rank approximations under Frobenius norm, such as minimizing the Frobenius norm under nuclear norm constraint, spectral norm constraint, orthogonality constraint and more. The geometric interpretation of the proposed approximation process along with its optimality for convex constraints is also discussed. In addition, it is shown how the approximation algorithm can be used for matrix completion as well, under a variety of spectral regularizations. Its applications to physics, electrical engineering and data interpolation problems are also described.
KW - Matrix approximation
KW - Matrix completion
UR - http://www.scopus.com/inward/record.url?scp=84908247467&partnerID=8YFLogxK
U2 - 10.13001/1081-3810.1551
DO - 10.13001/1081-3810.1551
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AN - SCOPUS:84908247467
VL - 23
SP - 678
EP - 702
JO - Electronic Journal of Linear Algebra
JF - Electronic Journal of Linear Algebra
SN - 1081-3810
M1 - 50
ER -