• Corpus ID: 16128140

A direct proof for Lovett's bound on the communication complexity of low rank matrices

@article{Rothvoss2014ADP,
  title={A direct proof for Lovett's bound on the communication complexity of low rank matrices},
  author={Thomas Rothvoss},
  journal={ArXiv},
  year={2014},
  volume={abs/1409.6366}
}
  • T. Rothvoss
  • Published 22 September 2014
  • Computer Science
  • ArXiv
The log-rank conjecture in communication complexity suggests that the deterministic communication complexity of any Boolean rank-r function is bounded by polylog(r). Recently, major progress was made by Lovett who proved that the communication complexity is bounded by O(r^1/2 * log r). Lovett's proof is based on known estimates on the discrepancy of low-rank matrices. We give a simple, direct proof based on a hyperplane rounding argument that in our opinion sheds more light on the reason why a… 
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References

SHOWING 1-10 OF 13 REFERENCES
En Route to the Log-Rank Conjecture: New Reductions and Equivalent Formulations
We prove that several measures in communication complexity are equivalent, up to polynomial factors in the logarithm of the rank of the associated matrix: deterministic communication complexity,
Communication is bounded by root of rank
We prove that any total boolean function of rank r can be computed by a deterministic communication protocol of complexity O(√r · log(r)). Similarly, any graph whose adjacency matrix has rank r has
An Additive Combinatorics Approach Relating Rank to Communication Complexity
TLDR
This work proves that, assuming the Polynomial Freiman-Ruzsa (PFR) conjecture in additive combinatorics, there exists a universal constant c such that CC(M) ≤ c ·rank(M)/log rank(M), and improves the bounds on approximate duality assuming the PFR conjecture.
On rank vs. communication complexity
  • N. Nisan, A. Wigderson
  • Mathematics
    Proceedings 35th Annual Symposium on Foundations of Computer Science
  • 1994
TLDR
This paper concerns the open problem of Lovasz and Saks (1988) regarding the relationship between the communication complexity of a Boolean function and the rank of the associated matrix and proves two related theorems.
Rank and chromatic number of a graph
It was proved (A. Kotlov and L. Lovasz, The rank and size of graphs, J. Graph Theory23(1996), 185–189) that the number of vertices in a twin-free graph is where r is the rank of the adjacency matrix.
Communication Complexity
TLDR
This work will review the basic communication model and some of the classical results known for it, sometimes even with proofs, and consider a variant in which the players are allowed to flip fair unbiased coins.
Complexity measures of sign matrices
TLDR
Four previously known parameters of sign matrices from a complexity-theoretic perspective are considered and tight (or nearly tight) inequalities that are established among these parameters are established.
Extremum Problems with Inequalities as Subsidiary Conditions
This paper deals with an extension of Lagrange’s multiplier rule to the case, where the subsidiary conditions are inequalities instead of equations. Only extrema of differentiable functions of a
Lattices, mobius functions and communications complexity
  • L. Lovász, M. Saks
  • Computer Science
    [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
  • 1988
A general framework for the study of a broad class of communication problems is developed. It is based on a recent analysis of the communication complexity of graph connectivity. The approach makes
The rank and size of graphs
We show that the number of points with pairwise different sets of neighbors in a graph is O(2r/2), where r is the rank of the adjacency matrix. We also give an example that achieves this bound. ©
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