×

Duality mapping for Schatten matrix norms. (English) Zbl 1484.15024

The paper focuses on the duality mapping defined on the vector space of \(m\times n\) matrices endowed with the Schatten norm. For \(1\leq p\leq\infty\), the Schatten \(p\)-norm of a matrix \(A\in\mathbb{R}^{m\times n}\) is defined as \[ \|A\|_{S_p}=\begin{cases}\left(\sum_{i=1}^r\sigma_i^p\right)^{1/p}& \text{if } p<\infty,\\ \sigma_1&\text{if }p=\infty,\end{cases} \] where \(r\) denotes the rank and \(\sigma_i\) (\(i=1,\dots,r\)) denote the singular values of \(A\). We say that the pair \((A,B)\in \mathbb{R}^{m\times n}\times \mathbb{R}^{m\times n}\) is \((\|\cdot\|_{S_p},\|\cdot\|_{S_q})\) conjugate if
\(1/p+1/q=1\);
\(\mathrm{Tr}(B^TA)=\|B\|_{S_q}\|A\|_{S_p}\);
\(\|B\|_{S_q}=\|A\|_{S_p}\).
Then the set-valued mapping \(\mathcal{J}\colon \mathbb{R}^{m\times n}\to 2^{\mathbb{R}^{m\times n}}\) is defined as follows: for any \(A\in \mathbb{R}^{m\times n}\to \mathbb{R}^{m\times n}\) let \(\mathcal{J}_{S_p}(A)\) denote the set of \(B\in\mathbb{R}^{m\times n}\) such that \((A,B)\) forms a \((\|\cdot\|_{S_p},\|\cdot\|_{S_q})\) conjugate pair. The main result of the paper is an explicit characterization of the duality mapping \(\mathcal{J}_{S_p}\). In fact, it turns out that for \(1<p<\infty\) this map is single-valued.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A45 Miscellaneous inequalities involving matrices
47A30 Norms (inequalities, more than one norm, etc.) of linear operators

Software:

PhaseLift
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bhatia, R., Matrix Analysis, 169 (1997), New York: Springer-Verlag, New York
[2] Lefkimmiatis, S.; Unser, M., Poisson image reconstruction with Hessian Schatten-norm regularization, IEEE Trans. Image Process, 22, 11, 4314-4327 (2013) · Zbl 1373.94228 · doi:10.1109/TIP.2013.2271852
[3] Lefkimmiatis, S.; Ward, J.; Unser, M., Hessian Schatten-norm regularization for linear inverse problems, IEEE Trans. Image Process, 22, 5, 1873-1888 (2013) · Zbl 1373.94229 · doi:10.1109/TIP.2013.2237919
[4] Xie, Y.; Gu, S.; Liu, Y.; Zuo, W.; Zhang, W.; Zhang, L., Weighted Schatten p-norm minimization for image denoising and background subtraction, IEEE Trans. Image Process, 25, 10, 4842-4857 (2016) · Zbl 1408.94731 · doi:10.1109/TIP.2016.2599290
[5] Gao, S.; Fan, Q., Robust Schatten-p norm based approach for tensor completion, J. Sci. Comput, 82, 1, 23 (2020) · Zbl 1436.90103
[6] Horn, R. A.; Johnson, C. R., Matrix Analysis (2012), Cambridge University Press
[7] Davenport, M. A.; Romberg, J., An overview of low-rank matrix recovery from incomplete observations, IEEE J. Sel. Top. Signal Process, 10, 4, 608-622 (2016) · doi:10.1109/JSTSP.2016.2539100
[8] Candès, E. J.; Recht, B., Exact matrix completion via convex optimization, Found. Comput. Math, 9, 6, 717-772 (2009) · Zbl 1219.90124 · doi:10.1007/s10208-009-9045-5
[9] Candès, E. J.; Eldar, Y. C.; Strohmer, T.; Voroninski, V., Phase retrieval via matrix completion, SIAM Rev, 57, 2, 225-251 (2015) · Zbl 1344.49057 · doi:10.1137/151005099
[10] Davies, M. E.; Eldar, Y. C., Rank awareness in joint sparse recovery, IEEE Trans. Inform. Theory, 58, 2, 1135-1146 (2012) · Zbl 1365.94175 · doi:10.1109/TIT.2011.2173722
[11] Fazel, M.; Pong, T. K.; Sun, D.; Tseng, P., Hankel matrix rank minimization with applications to system identification and realization, SIAM J. Matrix Anal. Appl, 34, 3, 946-977 (2013) · Zbl 1302.90127 · doi:10.1137/110853996
[12] Asadi, E.; Aziznejad, S.; Amerimehr, M. H.; Amini, A., A fast matrix completion method for index coding, 2606-2610 (2017), Kos Island, Greece: IEEE, Kos Island, Greece
[13] Esfahanizadeh, H.; Lahouti, F.; Hassibi, B., A matrix completion approach to linear index coding problem, 531-535 (2014), Hobart, Australia: IEEE, Hobart, Australia
[14] Kittaneh, F., Inequalities for the Schatten p-norm, Glasgow Math. J, 26, 2, 141-143 (1985) · Zbl 0578.47005 · doi:10.1017/S0017089500005905
[15] Kittaneh, F., Inequalities for the Schatten p-norm II, Glasgow Math. J, 29, 1, 99-104 (1987) · Zbl 0638.47004 · doi:10.1017/S0017089500006716
[16] Kittaneh, F., Inequalities for the Schatten p-norm III, Communmath. Phys, 104, 2, 307-310 (1986) · Zbl 0595.47012 · doi:10.1007/BF01211597
[17] Kittaneh, F., Inequalities for the Schatten p-norm IV, Communmath. Phys, 106, 4, 581-585 (1986) · Zbl 0612.47018 · doi:10.1007/BF01463397
[18] Kittaneh, F.; Kosaki, H., Inequalities for the Schatten p-norm V, Publ. Res. Inst. Math. Sci, 23, 2, 433-443 (1987) · Zbl 0627.47002 · doi:10.2977/prims/1195176547
[19] Bourin, J.-C., Matrix versions of some classical inequalities, Linear Algebra Appl, 416, 2-3, 890-907 (2006) · Zbl 1100.15010 · doi:10.1016/j.laa.2006.01.002
[20] Hirzallah, O.; Kittaneh, F.; Moslehian, M., Schatten p-norm inequalities related to a characterization of inner product spaces, Math. Inequal. Appl, 13, 2, 235-241 (2010) · Zbl 1201.47013 · doi:10.7153/mia-13-19
[21] Moslehian, M. S.; Tominaga, M.; Saito, K.-S., Schatten p-norm inequalities related to an extended operator parallelogram law, Linear Algebra Appl, 435, 4, 823-829 (2011) · Zbl 1218.47032 · doi:10.1016/j.laa.2011.01.046
[22] Conde, C.; Moslehian, M. S., Norm inequalities related to p-Schatten class, Linear Algebra Appl, 498, 441-449 (2016) · Zbl 1341.47011 · doi:10.1016/j.laa.2015.11.031
[23] Wenzel, D.; Audenaert, K. M., Impressions of convexity: An illustration for commutator bounds, Linear Algebra Appl, 433, 11-12, 1726-1759 (2010) · Zbl 1203.15013 · doi:10.1016/j.laa.2010.06.039
[24] Cheng, C.-M.; Lei, C., On Schatten p-norms of commutators, Linear Algebra Appl, 484, 409-434 (2015) · Zbl 1325.15017 · doi:10.1016/j.laa.2015.07.009
[25] So, W., Facial structures of Schatten p-norms, Linear Multilinear Algebra, 27, 3, 207-212 (1990) · Zbl 0706.15027 · doi:10.1080/03081089008818012
[26] Potapov, D.; Sukochev, F., Fréchet differentiability of \(####\) norms, Adv. Math, 262, 436-475 (2014) · Zbl 1311.46043
[27] Kittaneh, F., On the continuity of the absolute value map in the Schatten classes, Linear Algebra Appl, 118, 61-68 (1989) · Zbl 0676.47007 · doi:10.1016/0024-3795(89)90571-5
[28] Bhatia, R.; Kittaneh, F., Cartesian decompositions and Schatten norms, Linear Algebra Appl, 318, 1-3, 109-116 (2000) · Zbl 0981.47008 · doi:10.1016/S0024-3795(00)00206-8
[29] Beurling, A.; Livingston, A., A theorem on duality mappings in Banach spaces, Ark. Mat, 4, 5, 405-411 (1962) · Zbl 0105.09301 · doi:10.1007/BF02591622
[30] Cioranescu, I., Geometry of Banach Spaces. Duality Mappings and Nonlinear Problems, 62 (1990), Netherlands: Springer, Netherlands · Zbl 0712.47043
[31] de Boor, C., On “best” interpolation, J. Approximation Theory, 16, 1, 28-42 (1976) · Zbl 0314.41001 · doi:10.1016/0021-9045(76)90093-9
[32] Unser, M., A Unifying Representer Theorem for Inverse Problems and Machine Learning (2020), Foundations of Computational Mathematics
[33] Liu, P.; Wang, Y.-W., The best generalized inverse of the linear operator in normed linear space, Linear Algebra Appl, 420, 1, 9-19 (2007) · Zbl 1114.47001 · doi:10.1016/j.laa.2006.04.024
[34] Rudin, W., Functional Analysis. International Series in Pure and Applied Mathematics (1991), New York: McGraw-Hill, Inc, New York · Zbl 0867.46001
[35] Johnson, C. R.; Horn, R. A., Matrix Analysis (1985), Cambridge University Press · Zbl 0576.15001
[36] Nie, F.; Huang, H.; Ding, C., Low-rank matrix recovery via efficient Schatten p-norm minimization, 26 (2012)
[37] Shang, F.; Liu, Y.; Cheng, J., Scalable algorithms for tractable Schatten quasi-norm minimization, 30 (2016)
[38] Shang, F.; Liu, Y.; Shang, F.; Liu, H.; Kong, L.; Jiao, L., A unified scalable equivalent formulation for schatten quasi-norms, Mathematics, 8, 8, 1325 (2020) · doi:10.3390/math8081325
[39] Giampouras, P., Vidal, R., Rontogiannis, A., Haeffele, B. (2020). A novel variational form of the Schatten-p quasi-norm. arXiv preprint arXiv:2010.13927.
[40] Lefkimmiatis, S.; Roussos, A.; Maragos, P.; Unser, M., Structure tensor total variation, SIAM J. Imaging Sci, 8, 2, 1090-1122 (2015) · Zbl 1315.65019 · doi:10.1137/14098154X
[41] Petryshyn, W., A characterization of strict convexity of Banach spaces and other uses of duality mappings, J. Funct. Anal, 6, 2, 282-291 (1970) · Zbl 0199.44004 · doi:10.1016/0022-1236(70)90061-3
[42] Giles, J.; Gregory, D.; Sims, B., Geometrical implications of upper semi-continuity of the duality mapping on a Banach space, Pacific J. Math, 79, 1, 99-109 (1978) · Zbl 0399.46012 · doi:10.2140/pjm.1978.79.99
[43] Contreras, M. D.; Payá, R., On upper semicontinuity of duality mappings, Proc. Am. Math. Soc, 121, 2, 451-459 (1994) · Zbl 0818.46017 · doi:10.1090/S0002-9939-1994-1215199-4
[44] Himmelberg, C. J.; Parthasarathy, T., Measurable relations, Fund. Math, 87, 1, 53-72 (1975) · Zbl 0296.28003 · doi:10.4064/fm-87-1-53-72
[45] Cvetkovski, Z., Inequalities: Theorems, Techniques and Selected Problems (2012), Berlin Heidelberg: Springer-Verlag, Berlin Heidelberg · Zbl 1233.00003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.