Sampling on the Sphere by Mutually Orthogonal Subspaces

TL;DR

This paper establishes an $oldsymbol{ ilde{ ext{O}}}(oldsymbol{ ext{sqrt}}{n})$ lower bound for the Vector in Subspace Communication Problem with bounded rank, and proposes a conjecture linking measure concentration to protocol complexity.

Contribution

It provides an optimal lower bound for a specific communication problem and introduces a conjecture connecting measure concentration phenomena to protocol complexity.

Findings

Proves an $ ilde{O}( ext{sqrt}(n))$ lower bound for bounded rank protocols.

Verifies the conjecture for sets depending on $O( ext{sqrt}(n))$) directions.

Complements Raz's existing $O( ext{sqrt}(n))$ protocol with a matching lower bound.

Abstract

The purpose of this paper is twofold. First, we provide an optimal $\Omega(\sqrt{n})$ bits lower bound for any two-way protocol for the Vector in Subspace Communication Problem which is of bounded total rank. This result complements Raz's $O(\sqrt{n})$ protocol, which has a simple variant of bounded total rank. Second, we present a plausible mathematical conjecture on a measure concentration phenomenon that implies an $\Omega(\sqrt{n})$ lower bound for a general protocol. We prove the conjecture for the subclass of sets that depend only on $O(\sqrt{n})$ directions.