Communication Complexity of Inner Product in Symmetric Normed Spaces

by   Alexandr Andoni, et al.

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 ℝ^n. Here, Alice and Bob hold two vectors v,u such that v_N≤ 1 and u_N^*≤ 1, where N^* is the dual norm. They want to compute their inner product ⟨ v,u ⟩ up to an ε additive term. The problem is denoted by IP_N. We systematically study IP_N, showing the following results: - For any symmetric norm N, given v_N≤ 1 and u_N^*≤ 1 there is a randomized protocol for IP_N using 𝒪̃(ε^-6log n) bits – we will denote this by ℛ_ε,1/3(IP_N) ≤𝒪̃(ε^-6log n). - One way communication complexity ℛ(IP_ℓ_p)≤𝒪(ε^-max(2,p)·logn/ε), and a nearly matching lower bound ℛ(IP_ℓ_p) ≥Ω(ε^-max(2,p)) for ε^-max(2,p)≪ n. - One way communication complexity ℛ(N) for a symmetric norm N is governed by embeddings ℓ_∞^k into N. Specifically, while a small distortion embedding easily implies a lower bound Ω(k), we show that, conversely, non-existence of such an embedding implies protocol with communication k^𝒪(loglog k)log^2 n. - For arbitrary origin symmetric convex polytope P, we show ℛ(IP_N) ≤𝒪(ε^-2logxc(P)), where N is the unique norm for which P is a unit ball, and xc(P) is the extension complexity of P.


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