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Preserving positivity for matrices with sparsity constraints


Authors: Dominique Guillot, Apoorva Khare and Bala Rajaratnam
Journal: Trans. Amer. Math. Soc. 368 (2016), 8929-8953
MSC (2010): Primary 15B48; Secondary 26E05, 05C50, 26A48
DOI: https://doi.org/10.1090/tran6669
Published electronically: January 6, 2016
MathSciNet review: 3551594
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Abstract: Functions preserving Loewner positivity when applied entrywise to positive semidefinite matrices have been widely studied in the literature. Following the work of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], and others, it is well-known that functions preserving positivity for matrices of all dimensions are absolutely monotonic (i.e., analytic with nonnegative Taylor coefficients). In this paper, we study functions preserving positivity when applied entrywise to sparse matrices, with zeros encoded by a graph $ G$ or a family of graphs $ G_n$. Our results generalize Schoenberg and Rudin's results to a modern setting, where functions are often applied entrywise to sparse matrices in order to improve their properties (e.g. better conditioning, graphical models). The only such result known in the literature is for the complete graph $ K_2$. We provide the first such characterization result for a large family of non-complete graphs. Specifically, we characterize functions preserving Loewner positivity on matrices with zeros according to a tree. These functions are multiplicatively midpoint-convex and superadditive. Leveraging the underlying sparsity in matrices thus admits the use of functions which are not necessarily analytic nor absolutely monotonic. We further show that analytic functions preserving positivity on matrices with zeros according to trees can contain arbitrarily long sequences of negative coefficients, thus obviating the need for absolute monotonicity in a very strong sense. This result leads to the question of exactly when absolute monotonicity is necessary when preserving positivity for an arbitrary class of graphs. We then provide a stronger condition in terms of the numerical range of all symmetric matrices, such that functions satisfying this condition on matrices with zeros according to any family of graphs with unbounded degrees are necessarily absolutely monotonic.


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Additional Information

Dominique Guillot
Affiliation: Departments of Mathematics and Statistics, Stanford University, Stanford, California 94305
Address at time of publication: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
Email: dguillot@udel.edu

Apoorva Khare
Affiliation: Departments of Mathematics and Statistics, Stanford University, Stanford, California 94305
Email: khare@stanford.edu

Bala Rajaratnam
Affiliation: Departments of Mathematics and Statistics, Stanford University, Stanford, California 94305
Email: brajaratnam01@gmail.com

DOI: https://doi.org/10.1090/tran6669
Keywords: Matrices with structure of zeros, entrywise positive maps, absolutely monotonic functions, multiplicatively convex functions, positive semidefiniteness, Loewner ordering, fractional Schur powers
Received by editor(s): November 27, 2014
Published electronically: January 6, 2016
Additional Notes: This work was partially supported by US Air Force Office of Scientific Research grant award FA9550-13-1-0043, US National Science Foundation under grant DMS-0906392, DMS-CMG 1025465, AGS-1003823, DMS-1106642, DMS-CAREER-1352656, Defense Advanced Research Projects Agency DARPA YFA N66001-111-4131, the UPS Foundation, SMC-DBNKY, and an NSERC postdoctoral fellowship
Article copyright: © Copyright 2016 American Mathematical Society