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Sign-rank Can Increase under Intersection

Published:01 September 2021Publication History
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Abstract

The communication class UPPcc is a communication analog of the Turing Machine complexity class PP. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds.

For a communication problem f, let ff denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPPcc(f) = O(log n), and UPPcc(ff) = Θ (log2 n). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPPcc, the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection.

Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity nOmegaΩ(log n). This matches an upper bound of (Klivans, O’Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time.

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 13, Issue 4
        December 2021
        198 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3481683
        Issue’s Table of Contents

        Copyright © 2021 Association for Computing Machinery.

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 1 September 2021
        • Accepted: 1 April 2021
        • Revised: 1 February 2021
        • Received: 1 May 2020
        Published in toct Volume 13, Issue 4

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