The aBc Problem and Equator Sampling Renyi Divergences
Abstract
We investigate the problem of approximating the product $a^TBc$, where $a,c\in S^{n1}$ and $B\in O_n$, in models of communication complexity and streaming algorithms. The worst meaningful approximation is to simply decide whether the product is 1 or 1, given the promise that it is either. We call that problem the aBc problem. This is a modification of computing approximate inner products, by allowing a basis change. While very efficient streaming algorithms and oneway communication protocols are known for simple inner products (approximating $a^Tc$) we show that no efficient oneway protocols/streaming algorithms exist for the aBc problem. In communication complexity we consider the 3player numberinhand model. 1) In communication complexity $a^TBc$ can be approximated within additive error $\epsilon$ with communication $O(\sqrt n/\epsilon^2)$ by a oneway protocol Charlie to Bob to Alice. 2) The $aBc$ problem has a streaming algorithm that uses space $O(\sqrt n \log n)$. 3) Any oneway communication protocol for $aBc$ needs communication at least $\Omega(n^{1/3})$, and we prove a tight results regarding a communication tradeoff: if Charlie and Bob communicate over many rounds such that Charlie communicates $o(n^{2/3})$ and Bob $o(n^{1/3})$, and then the transcript is sent to Alice, the error will be large. 4) To establish our lower bound we show concentration results for Renyi divergences under the event of restricting a density function on the sphere to a random equator and subsequently normalizing the restricted density function. This extends previous results by Klartag and Regev for set sizes to Renyi divergences of arbitrary density functions. 5) We show a strong concentration result for conditional Renyi divergences on bipartite systems for all $\alpha>1$.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 arXiv:
 arXiv:1912.11275
 Bibcode:
 2019arXiv191211275K
 Keywords:

 Computer Science  Computational Complexity;
 Quantum Physics