## 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 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. 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^{T}Av 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 ^{2}_{3} -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)} log^{2} 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^{′}).

Original language | English |
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Title of host publication | 14th Innovations in Theoretical Computer Science Conference, ITCS 2023 |

Editors | Yael Tauman Kalai |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772631 |

DOIs | |

State | Published - 1 Jan 2023 |

Event | 14th Innovations in Theoretical Computer Science Conference, ITCS 2023 - Cambridge, United States Duration: 10 Jan 2023 → 13 Jan 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 251 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 14th Innovations in Theoretical Computer Science Conference, ITCS 2023 |
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Country/Territory | United States |

City | Cambridge |

Period | 10/01/23 → 13/01/23 |

### Bibliographical note

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

## Keywords

- communication complexity
- symmetric norms