TY - JOUR

T1 - Continuity of closest rank-p approximations to matrices

AU - Mittelmann, Hans

AU - Cadzow, James A.

PY - 1987/8

Y1 - 1987/8

N2 - In signal processing, the singular value decomposition and rank characterization of matrices play prominent roles. The mapping which associates with any complex m x n matrix X its closest rank-p approximation X(p)need not be continuous. When the pth and the (p + 1)st singular values of X are equal, this mapping maps, in fact, a matrix to a set of matrices. Furthermore, an example is given to show that large errors in computing X(p)can be expected when σ is sufficiently close to αP+ I,. It is finally shown that this mapping is closed in the sense of Zangwill. The property of closedness is an essential assumption of a global convergence proof for algorithms involving this mapping (e.g., see [l]).

AB - In signal processing, the singular value decomposition and rank characterization of matrices play prominent roles. The mapping which associates with any complex m x n matrix X its closest rank-p approximation X(p)need not be continuous. When the pth and the (p + 1)st singular values of X are equal, this mapping maps, in fact, a matrix to a set of matrices. Furthermore, an example is given to show that large errors in computing X(p)can be expected when σ is sufficiently close to αP+ I,. It is finally shown that this mapping is closed in the sense of Zangwill. The property of closedness is an essential assumption of a global convergence proof for algorithms involving this mapping (e.g., see [l]).

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U2 - 10.1109/TASSP.1987.1165262

DO - 10.1109/TASSP.1987.1165262

M3 - Article

AN - SCOPUS:0023401172

VL - 35

SP - 1211

EP - 1212

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

IS - 8

ER -