Nonsmooth Analysis of Singular Values. Part I: Theory |
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| Authors: | Adrian S Lewis and Hristo S Sendov |
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| Institution: | (1) Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada;(2) Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada |
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| Abstract: | The singular values of a rectangular matrix are nonsmooth functions of its entries. In this work we study the nonsmooth analysis of functions of singular values. In particular we give simple formulae for the regular subdifferential, the limiting subdifferential, and the horizon subdifferential, of such functions. Along the way to the main result we give several applications and in particular derive von Neumann’s trace inequality for singular values.
Mathematics Subject Classifications (2000) Primary 90C31, 15A18; secondary 49K40, 26B05.Research supported by NSERC. |
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| Keywords: | nonsmooth analysis singular values regular subdifferential limiting subdifferential horizon subdifferential von Neumann trace inequality simultaneous diagonalization |
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