# Communication Complexity of Inner Product in Symmetric Normed Spaces

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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