In an influential paper, Linial and Shraibman (STOC '07) introduced the factorization norm as a powerful tool for proving lower bounds against randomized and quantum communication complexities. They showed that the logarithm of the approximate γ₂-factorization norm is a lower bound for these parameters and asked whether a stronger lower bound that replaces approximate γ₂ norm with the γ₂ norm holds. We answer the question of Linial and Shraibman in the negative by exhibiting a 2ⁿ×2ⁿ Boolean matrix with γ₂ norm 2^Ω(n) and randomized communication complexity O(log n). As a corollary, we recover the recent result of Chattopadhyay, Lovett, and Vinyals (CCC '19) that deterministic protocols with access to an Equality oracle are exponentially weaker than (one-sided error) randomized protocols. In fact, as a stronger consequence, our result implies an exponential separation between the power of unambiguous nondeterministic protocols with access to Equality oracle and (one-sided error) randomized protocols, which answers a question of Pitassi, Shirley, and Shraibman (ITSC '23). Our result also implies a conjecture of Sherif (Ph.D. thesis) that the γ₂ norm of the Integer Inner Product function (IIP) in dimension 3 or higher is exponential in its input size.
@InProceedings{cheung_et_al:LIPIcs.CCC.2023.1, author = {Cheung, Tsun-Ming and Hatami, Hamed and Hosseini, Kaave and Shirley, Morgan}, title = {{Separation of the Factorization Norm and Randomized Communication Complexity}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {1:1--1:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.1}, URN = {urn:nbn:de:0030-drops-182714}, doi = {10.4230/LIPIcs.CCC.2023.1}, annote = {Keywords: Factorization norms, randomized communication complexity} }
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