The resolvent average for positive semidefinite matrices. (English) Zbl 1191.15024
Given positive semidefinite matrices \(A_1,\dots,A_n\) of the same size, and \(\lambda_1,\dots,\lambda_n\in (0,\infty)\) with \(\sum_{i=1}^n \lambda_i=1\), the authors define the resolvent average to be
\[ R_\mu(A,\lambda):= \big( \lambda_1 (A_1+\mu^{-1})^{-1} + \cdots + \lambda_n (A_n+\mu^{-1})^{-1} \big)^{-1} - \mu^{-1}. \]
Note \(R_1(A,\lambda)\) is a weighted average of resolvents of the \(A_i\). The authors study this notion and compare it to the well-known harmonic and arithmetic average defined as \( H(A,\lambda)= \big( \lambda_1 A_1^{-1} + \cdots + \lambda_n A_n^{-1} \big)^{-1}\) and \( A(A,\lambda)= \lambda_1 A_1 + \cdots + \lambda_n A_n\), respectively.
As a sample of the results obtained, we mention
\[ H(A,\lambda) \preceq R_\mu(A,\lambda) \preceq A(A,\lambda), \]
where \(\preceq\) stands for the Löwner partial ordering, and
\[ \lim_{\mu\searrow 0} R_\mu(A,\lambda) = A(A,\lambda), \qquad \lim_{\mu\to\infty} R_\mu(A,\lambda) = H(A,\lambda). \]
The paper is well-written, and the proofs are based on convexity arguments.
\[ R_\mu(A,\lambda):= \big( \lambda_1 (A_1+\mu^{-1})^{-1} + \cdots + \lambda_n (A_n+\mu^{-1})^{-1} \big)^{-1} - \mu^{-1}. \]
Note \(R_1(A,\lambda)\) is a weighted average of resolvents of the \(A_i\). The authors study this notion and compare it to the well-known harmonic and arithmetic average defined as \( H(A,\lambda)= \big( \lambda_1 A_1^{-1} + \cdots + \lambda_n A_n^{-1} \big)^{-1}\) and \( A(A,\lambda)= \lambda_1 A_1 + \cdots + \lambda_n A_n\), respectively.
As a sample of the results obtained, we mention
\[ H(A,\lambda) \preceq R_\mu(A,\lambda) \preceq A(A,\lambda), \]
where \(\preceq\) stands for the Löwner partial ordering, and
\[ \lim_{\mu\searrow 0} R_\mu(A,\lambda) = A(A,\lambda), \qquad \lim_{\mu\to\infty} R_\mu(A,\lambda) = H(A,\lambda). \]
The paper is well-written, and the proofs are based on convexity arguments.
Reviewer: Igor Klep (Ljubljana)
MSC:
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 15A45 | Miscellaneous inequalities involving matrices |
| 47A64 | Operator means involving linear operators, shorted linear operators, etc. |
| 52A41 | Convex functions and convex programs in convex geometry |
Keywords:
positive semidefinite matrix; arithmetic mean; harmonic mean; arithmetic average; convex function; Fenchel conjugate; harmonic average; proximal average; resolvent average; subdifferential; Löwner partial orderingReferences:
| [1] | Anderson, W. N.; Duffin, R. J., Series and parallel addition of matrices, J. Math. Anal. Appl., 26, 576-594 (1969) · Zbl 0177.04904 |
| [2] | Ando, T.; Li, C.-K.; Mathias, R., Geometric means, Linear Algebra Appl., 385, 305-334 (2004) · Zbl 1063.47013 |
| [3] | Bauschke, H. H.; Borwein, J. M., Legendre functions and the method of random Bregman projections, J. Convex Anal., 4, 27-67 (1997) · Zbl 0894.49019 |
| [4] | Bauschke, H. H.; Matoušková, E.; Reich, S., Projection and proximal point methods: convergence results and counterexamples, Nonlinear Anal., 56, 715-738 (2004) · Zbl 1059.47060 |
| [5] | H.H. Bauschke, S.M. Moffat, X. Wang, Resolvent averages of monotone operators, preprint.; H.H. Bauschke, S.M. Moffat, X. Wang, Resolvent averages of monotone operators, preprint. |
| [6] | Bauschke, H. H.; Wang, X., The kernel average for two convex functions and its applications to the extension and representation of monotone operators, Trans. Amer. Math. Soc., 361, 5947-5965 (2009) · Zbl 1189.47051 |
| [7] | Bauschke, H. H.; Goebel, R.; Lucet, Y.; Wang, X., The proximal average: basic theory, SIAM J. Optim., 19, 766-785 (2008) · Zbl 1172.26003 |
| [8] | Bauschke, H. H.; Lucet, Y.; Trienis, M., How to transform one convex function continuously into another, SIAM Rev., 50, 115-132 (2008) · Zbl 1145.90050 |
| [9] | Bauschke, H. H.; Lucet, Y.; Wang, X., Primal-dual symmetric intrinsic methods for finding antiderivatives of cyclically monotone operators, SIAM J. Control Optim., 46, 2031-2051 (2007) · Zbl 1189.47050 |
| [10] | Bhatia, R.; Holbrook, J., Riemannian geometry and matrix geometric means, Linear Algebra Appl., 413, 594-618 (2006) · Zbl 1088.15022 |
| [11] | Borwein, J. M.; Lewis, A. S., Convex Analysis and Nonlinear Optimization (2000), Springer-Verlag: Springer-Verlag New York · Zbl 0953.90001 |
| [12] | Boyd, S.; Vandenberghe, L., Convex Optimization (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1058.90049 |
| [13] | Hiriart-Urruty, J.-B.; Mazure, M.-L., Formulations variationnelles de l’addition parallèle et de la soustraction parallèlle d’opérateurs semi-définis positifs, C.R. Acad. Sci. Paris Sér. I Math., 302, 527-530 (1986) · Zbl 0597.15008 |
| [14] | Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press · Zbl 0576.15001 |
| [15] | Kubo, F.; Ando, T., Means of positive linear operators, Math. Ann., 246, 3, 205-224 (1980) · Zbl 0412.47013 |
| [16] | Marshall, A. W.; Olkin, I., Inequalities: Theory of Majorization and Its Applications (1979), Academic Press: Academic Press San Diego, CA · Zbl 0437.26007 |
| [17] | Mazure, M.-L., L’addition parallèle d’opérateurs interprétée comme inf-convolution de formes quadratiques convexes, RAIRO Modél. Math. Anal. Numér., 20, 497-515 (1986) · Zbl 0599.94010 |
| [18] | Minty, G. J., Monotone (nonlinear) operators in Hilbert space, Duke Math. J., 29, 341-346 (1962) · Zbl 0111.31202 |
| [19] | Moakher, M., A differential geometric approach to the geometric mean of symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl., 26, 735-747 (2005) · Zbl 1079.47021 |
| [20] | Moreau, J.-J., Proximité et dualité dans un espace hilbertien, Bull. Soc. Math. France, 93, 273-299 (1965) · Zbl 0136.12101 |
| [21] | Passty, G. B., The parallel sum of nonlinear monotone operators, Nonlinear Anal., 10, 215-227 (1986) · Zbl 0628.47033 |
| [22] | Petz, D., Means of positive matrices: geometry and a conjecture, Ann. Math. Inform., 32, 129-139 (2005) · Zbl 1115.47018 |
| [23] | Rockafellar, R. T., Convex Analysis (1970), Princeton University Press · Zbl 0229.90020 |
| [24] | Rockafellar, R. T.; Wets, R. J.-B., Variational Analysis (1998), Springer-Verlag · Zbl 0888.49001 |
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.
