Communication Complexity, Corner-Free Sets and the Symmetric Subrank of Tensors

by   Matthias Christandl, et al.

We develop and apply new combinatorial and algebraic tools to understand multiparty communication complexity in the Number On the Forehead (NOF) model, and related Ramsey type problems in combinatorics. We identify barriers for progress and propose new techniques to circumvent these. (1) We introduce a technique for constructing independent sets in hypergraphs via combinatorial degeneration. In particular, we make progress on the corner problem by proving the existence of a corner-free subset of 𝔽_2^n ×𝔽_2^n of size 3.16^n-o(n), which improves the previous lower bound 2.82^n of Linial, Pitassi and Shraibman (ITCS 2018). In the Eval problem over a group G, three players need to determine whether their inputs x_1, x_2, x_3 ∈ G sum to zero. As a consequence of our construction of corner-free sets, the communication complexity of the Eval problem over 𝔽_2^n is at most 0.34n + O(log n), which improves the previous upper bound 0.5n + O(log n). (2) We point out how induced matchings in hypergraphs pose a barrier for existing tensor tools (like slice rank, subrank, analytic rank, geometric rank and G-stable rank) to effectively upper bound the size of independent sets in hypergraphs. This implies a barrier for these tools to effectively lower bound the communication complexity of the Eval problem over any group G. (3) We introduce the symmetric subrank of tensors as a proposal to circumvent the induced matching barrier and we introduce the symmetric quantum functional as a symmetric variation on the quantum functionals (STOC 2018). We prove that Comon's conjecture holds asymptotically for the tensor rank, the subrank and the restriction preorder, which implies a strong connection between Strassen's asymptotic spectrum of tensors and the asymptotic spectrum of symmetric tensors.



There are no comments yet.


page 1

page 2

page 3

page 4

βˆ™ 11/16/2021

Larger Corner-Free Sets from Combinatorial Degenerations

There is a large and important collection of Ramsey-type combinatorial p...
βˆ™ 09/22/2017

Universal points in the asymptotic spectrum of tensors

The asymptotic restriction problem for tensors is to decide, given tenso...
βˆ™ 05/08/2019

The asymptotic induced matching number of hypergraphs: balanced binary strings

We compute the asymptotic induced matching number of the k-partite k-uni...
βˆ™ 12/17/2018

Barriers for fast matrix multiplication from irreversibility

The determination of the asymptotic algebraic complexity of matrix multi...
βˆ™ 06/03/2020

On tensor rank and commuting matrices

Obtaining superlinear lower bounds on tensor rank is a major open proble...
βˆ™ 11/06/2019

Variety Membership Testing, Algebraic Natural Proofs, and Geometric Complexity Theory

We study the variety membership testing problem in the case when the var...
βˆ™ 04/08/2019

More barriers for rank methods, via a "numeric to symbolic" transfer

We prove new barrier results in arithmetic complexity theory, showing se...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.