Duality Mapping for Schatten Matrix Norms. (15th June 2021)

• Record Type: Journal Article
• Authors: Aziznejad, Shayan; Unser, Michael
• Is Part Of: Numerical functional analysis and optimization. Volume 42:Number 6(2021)
• Page Start: 679
• Page End: 695
• Publication Date: 2021-06-15
• Subjects: Banach spaces -- duality mapping -- Hölder inequality -- Schatten norm -- singular value decomposition; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2021.1922438
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: In this paper, we fully characterize the duality mapping over the space of matrices that are equipped with Schatten norms. Our approach is based on the analysis of the saturation of the Hölder inequality for Schatten norms. We prove in our main result that, for p ∈ ( 1, ∞ ), the duality mapping over the space of real-valued matrices with Schatten- p norm is a continuous and single-valued function and provide an explicit form for its computation. For the special case p = 1, the mapping is set-valued; by adding a rank constraint, we show that it can be reduced to a Borel-measurable single-valued function for which we also provide a closed-form expression.