Communication Complexity of Inner Product in Symmetric Normed Spaces

Abstract

We introduce and study the communication complexity of computing the inner product of two vectors, where the input is restricted w.r.t. a norm $N$ on the space ${\mathbb{R}}^n$. Here, Alice and Bob hold two vectors $v,u$ such that $\Vert v{\Vert }_N\le 1$ and $\Vert u{\Vert }_{N^{\ast }}\le 1$, where $N^{\ast }$ is the dual norm. They want to compute their inner product $⟨v,u⟩$ up to an $\epsilon$ additive term. The problem is denoted by ${\mathrm{IP}}_N$. We systematically study ${\mathrm{IP}}_N$, showing the following results: - For any symmetric norm $N$, given $\Vert v{\Vert }_N\le 1$ and $\Vert u{\Vert }_{N^{\ast }}\le 1$ there is a randomized protocol for ${\mathrm{IP}}_N$ using $\tilde{\mathcal{O}}({\epsilon }^{-6}\log n)$ bits -- we will denote this by ${\mathcal{R}}_{\epsilon ,1/3}({\mathrm{IP}}_N)\le \tilde{\mathcal{O}}({\epsilon }^{-6}\log n)$. - One way communication complexity $\overrightarrow{\mathcal{R}}({\mathrm{IP}}_{{\ell }_p})\le \mathcal{O}({\epsilon }^{-\max (2,p)}\cdot \log \frac{n}{\epsilon })$, and a nearly matching lower bound $\overrightarrow{\mathcal{R}}({\mathrm{IP}}_{{\ell }_p})\ge \Omega ({\epsilon }^{-\max (2,p)})$ for ${\epsilon }^{-\max (2,p)}\ll n$. - One way communication complexity $\overrightarrow{\mathcal{R}}(N)$ for a symmetric norm $N$ is governed by embeddings ${\ell }_{\infty }^k$ into $N$. Specifically, while a small distortion embedding easily implies a lower bound $\Omega (k)$, we show that, conversely, non-existence of such an embedding implies protocol with communication $k^{\mathcal{O}(\log \log k)}{\log }^{2}n$. - For arbitrary origin symmetric convex polytope $P$, we show $\mathcal{R}({\mathrm{IP}}_N)\le \mathcal{O}({\epsilon }^{-2}\log \mathrm{xc}(P))$, where $N$ is the unique norm for which $P$ is a unit ball, and $\mathrm{xc}(P)$ is the extension complexity of $P$.

Accepted to ITCS 2023