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Continuity of closest rank-p approximations to matrices

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2 Author(s)
Mittelmann, H. ; Arizona State University, Tempe, AZ ; Cadzow, James A.

In signal processing, the singular value decomposition and rank characterization of matrices play prominent roles. The mapping which associates with any complex m × 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 σpis sufficiently close to σp+1. 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 [1]).

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Acoustics, Speech and Signal Processing, IEEE Transactions on  (Volume:35 ,  Issue: 8 )