The aBc Problem and Equator Sampling Renyi Divergences

TL;DR

This paper studies the aBc problem, a variant of inner product approximation involving basis changes, and establishes bounds and concentration results for communication complexity and streaming algorithms, highlighting the problem's computational hardness.

Contribution

It introduces the aBc problem, proves new lower bounds for one-way communication protocols, and extends concentration results for Renyi divergences on spheres.

Findings

One-way protocols require at least (n^{1/3}) communication.

Streaming algorithms can approximate aBc within O((\u221a n ( ( space.

Concentration results for Renyi divergences are extended to arbitrary densities and bipartite systems.

Abstract

We investigate the problem of approximating the product $a^TBc$, where $a,c\in S^{n-1}$ 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 one-way communication protocols are known for simple inner products (approximating $a^Tc$) we show that no efficient one-way protocols/streaming algorithms exist for the aBc problem. In communication complexity we consider the 3-player number-in-hand model. 1) In communication complexity $a^TBc$ can be approximated within additive error $\epsilon$ with communication $O(\sqrt n/\epsilon^2)$ by a one-way protocol Charlie to Bob to Alice. 2) The $aBc$ problem has a streaming algorithm that uses space $O(\sqrt n \log n)$. 3) Any one-way 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 $\alpha>1$.