# 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} }

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…

## 9 Citations

CS 2429-Comm Complexity : Applications and New Directions Lecturer : Lalla Mouatadid 1 Additive Combinatorics and Log Rank

- Mathematics
- 2014

where the lower bound is due to Kushilevitz [unpublished, 1995] and the upper bound due to Kotlov in [5]. In 2012, Ben Sasson, Lovett and Zewi gave a conditional improvement on the upper bound [2],…

Bounded Matrix Rigidity and John's Theorem

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2016

Using John’s Theorem, we prove a lower bound on the bounded rigidity of a sign matrix, defined as the Hamming distance between this matrix and the set of low-rank, real-valued matrices with entries…

Exponential Separation between Quantum Communication and Logarithm of Approximate Rank

- Computer Science, Mathematics2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
- 2019

It is shown that even the quantum communication complexity of the sink function is polynomial, thus also refuting the quantum log-approximate-rank conjecture and giving a new proof of the classical lower bound using the fooling distribution method.

Quantum entanglement, sum of squares, and the log rank conjecture

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2017

The algorithm is based on the sum-of-squares hierarchy and its analysis is inspired by Lovett's proof that the communication complexity of every rank-n Boolean matrix is bounded by Õ(√n).

Point-hyperplane incidence geometry and the log-rank conjecture

- MathematicsACM Transactions on Computation Theory
- 2022

An unpublished conjecture of Golovnev, Meka, Sudan, and Velusamy on point-hyperplane incidence graphs, which posits the existence of large complete bipartite subgraphs (of density 2−polylog(d)) if the entire graph has constant density is exposit.

Near-Optimal Reviewer Splitting in Two-Phase Paper Reviewing and Conference Experiment Design

- Computer ScienceAAMAS
- 2022

It is shown both empirically (on real conference data) and theoretically (under certain natural conditions) that dividing reviewers uniformly at random is near-optimal.

Combinatorial Perspective on the Log-rank Conjecture

- Mathematics, Computer Science
- 2019

The log-rank conjecture states that the communication complexity of a 01-matrix is polylogarithmic in its rank, and an overview of results is given, focusing on the most recent upper bound by Lovett.

Upper Bounds on Communication in terms of Approximate Rank

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2019

It is shown that any Boolean function with approximate rank r can be computed by bounded error quantum protocols without prior entanglement of complexity O( √ r log r), and a strengthening of Newman's theorem with respect to approximate rank is obtained.

Technical Report Column

- Computer ScienceSIGA
- 2014

Simultaneous Approximation of Constraint Satisfaction Problems, Amey Bhangale, Swastik Kopparty, Sushant Sachdeva, TR14-098. The Complexity of DNF of Parities, Gil Cohen, Igor Shinkar, TR14-099. The…

## References

SHOWING 1-10 OF 13 REFERENCES

En Route to the Log-Rank Conjecture: New Reductions and Equivalent Formulations

- Computer Science, MathematicsICALP
- 2013

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

- Computer Science, MathematicsSTOC
- 2013

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

- Computer Science, Mathematics2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
- 2012

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

- MathematicsProceedings 35th Annual Symposium on Foundations of Computer Science
- 1994

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

- MathematicsJ. Graph Theory
- 1997

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

- Computer Science
- 2011

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

- MathematicsComb.
- 2007

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

- Mathematics
- 2014

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

- 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

- Mathematics, Computer ScienceJ. Graph Theory
- 1996

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. ©…