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 Rⁿ. Here, Alice and Bob hold two vectors v, u such that ∥v∥N ≤ 1 and ∥u∥N∗ ≤ 1, where N^∗ is the dual norm. The goal is to compute their inner product <v, u> up to an ε additive term. The problem is denoted by IPN, and generalizes important previously studied problems, such as: (1) Computing the expectation Ex∼D[f(x)] when Alice holds D and Bob holds f is equivalent to IP_ℓ1. (2) Computing v^TAv where Alice has a symmetric matrix with bounded operator norm (denoted S∞) and Bob has a vector v where ∥v∥2 = 1. This problem is complete for quantum communication complexity and is equivalent to IPS_∞. We systematically study IPN, showing the following results, near tight in most cases: 1. For any symmetric norm N, given ∥v∥N ≤ 1 and ∥u∥N∗ ≤ 1 there is a randomized protocol using Õ(ε⁻⁶ log n) bits of communication that returns a value in <u, v> ± ϵ with probability ²₃ -we will denote this by R_ε1/₃(IPN) ≤ Õ(ε⁻⁶ log n). In a special case where N = ℓp and N^∗ = ℓq for p⁻¹ + q⁻¹ = 1, we obtain an improved bound R_ε1/₃(IP_ℓp) ≤ O(ε⁻² log n), nearly matching the lower bound R_ε1/₃(IP_ℓp) ≥ Ω(min(n, ε⁻²)). 2. One way communication complexity ^−→R_ε,δ(IP_ℓp) ≤ O(ε^−max(2,p) · log ⁿ_ε ), and a nearly matching lower bound ^−→R_ε1/₃(IP_ℓp) ≥ Ω(ε^−max(2,p)) for ε^−max(2,p) ≪ n. 3. One way communication complexity ^−→R_ε,δ(N) for a symmetric norm N is governed by the distortion of the embedding 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(log log k)} log² n. 4. For arbitrary origin symmetric convex polytope P, we show R_ε1/₃(IPN) ≤ O(ε⁻² log xc(P)), where N is the unique norm for which P is a unit ball, and xc(P) is the extension complexity of P (i.e. the smallest number of inequalities describing some polytope P^′ s.t. P is projection of P^′).