Abstract
We provide an information-theoretic framework for establishing strong lower bounds on the nonnegative rank of matrices by means of common information, a notion previously introduced in Wyner (IEEE Trans Inf Theory 21(2):163–179, 1975). The framework is a generalization of the one in Braverman and Moitra (Proceedings of the forty-fifth annual ACM symposium on theory of computing, pp 161–170, 2013) for the shifted uniqe disjointness (UDISJ) matrix to arbitrary nonnegative matrices. Common information is a natural lower bound for the nonnegative rank of a matrix and by combining it with Hellinger distance estimations we compute the (almost) exact common information of UDISJ partial matrix. The bounds are obtained very naturally. We also establish robustness of this estimation under random or adversarial removal of rows and columns of the UDISJ partial matrix. This robustness translates, via a variant of Yannakakis’s factorization theorem, to lower bounds on the average case and adversarial approximate extension complexity of removals. We present the first family of polytopes, the hard pair introduced in Braun et al. (Math Oper Res 40(3):756–772, 2015) related to the CLIQUE problem, with high average case and adversarial approximate extension complexity of removals. The framework relies on a strengthened version of the link between information theory and Hellinger distance from Bar-Yossef (J Comput Syst Sci 68(4):702–732, 2004). We also provide an information theoretic variant of the fooling set method that allows us to extend fooling set lower bounds from extension complexity to approximate extension complexity.
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Acknowledgments
We are indebted to Samuel Fiorini for providing extensive feedback through the various stages of this work and who helped to significantly improve the presentation. We would also like to thank Daniel Dadush, Kostya Pashkovich, Santosh Vempala, and Ronald de Wolf for extremely valuable feedback as well as Jérémie Roland for pointing us to [22]. Part of this work was conducted at the Dagstuhl Seminar 13082 on Communication Complexity, Linear Optimization, and lower bounds for the nonnegative rank of matrices (http://www.dagstuhl.de/13082). The authors would like to thank the organizers for providing such a stimulating environment. Research reported in this paper was partially supported by NSF Grant CMMI-1300144.
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Braun, G., Pokutta, S. Common Information and Unique Disjointness. Algorithmica 76, 597–629 (2016). https://doi.org/10.1007/s00453-016-0132-0
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DOI: https://doi.org/10.1007/s00453-016-0132-0