Communication Complexity of Inner Product in Symmetric Normed Spaces

Alexandr Andoni, Jarosław Błasiok, Arnold Filtser

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations


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 IPℓ1. (2) Computing vTAv 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 Rε1/3(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 Rε1/3(IPℓp) ≤ O(ε−2 log n), nearly matching the lower bound Rε1/3(IPℓp) ≥ Ω(min(n, ε−2)). 2. One way communication complexity −→Rε,δ(IPℓp) ≤ O(ε−max(2,p) · log nε ), and a nearly matching lower bound −→Rε1/3(IPℓp) ≥ Ω(ε−max(2,p)) for ε−max(2,p) ≪ n. 3. One way communication complexity −→Rε,δ(N) for a symmetric norm N is governed by the distortion of the embedding ℓk 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 kO(log log k) log2 n. 4. For arbitrary origin symmetric convex polytope P, we show Rε1/3(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).

Original languageEnglish
Title of host publication14th Innovations in Theoretical Computer Science Conference, ITCS 2023
EditorsYael Tauman Kalai
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772631
StatePublished - 1 Jan 2023
Event14th Innovations in Theoretical Computer Science Conference, ITCS 2023 - Cambridge, United States
Duration: 10 Jan 202313 Jan 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference14th Innovations in Theoretical Computer Science Conference, ITCS 2023
Country/TerritoryUnited States

Bibliographical note

Publisher Copyright:
© Alexandr Andoni, Jarosław Błasiok, and Arnold Filtser; licensed under Creative Commons License CC-BY 4.0.


  • communication complexity
  • symmetric norms


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