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A Lower Bound on the Trace Norm of Boolean Matrices and Its Applications

Authors Tsun-Ming Cheung, Hamed Hatami , Kaave Hosseini , Aleksandar Nikolov, Toniann Pitassi, Morgan Shirley

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Author Details

Tsun-Ming Cheung
  • School of Computer Science, McGill University, Montreal, Canada
Hamed Hatami
  • School of Computer Science, McGill University, Montreal, Canada
Kaave Hosseini
  • Department of Computer Science, University of Rochester, NY, USA
Aleksandar Nikolov
  • Department of Computer Science, University of Toronto, Canada
Toniann Pitassi
  • Department of Computer Science, Columbia University, New York, NY, USA
Morgan Shirley
  • Department of Computer Science, University of Victoria, Canada

Funding

  • Hatami, Hamed: Supported by an NSERC grant.
  • Hosseini, Kaave: Partially supported by the Goergen Institute for Data Science at the University of Rochester.
  • Shirley, Morgan: Research was done while the author was a student at the University of Toronto. Supported by an NSERC grant.

Cite As Get BibTex

Tsun-Ming Cheung, Hamed Hatami, Kaave Hosseini, Aleksandar Nikolov, Toniann Pitassi, and Morgan Shirley. A Lower Bound on the Trace Norm of Boolean Matrices and Its Applications. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 37:1-37:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.37

Abstract

We present a simple method based on a variant of Hölder’s inequality to lower-bound the trace norm of Boolean matrices. As the main result, we obtain an exponential separation between the randomized decision tree depth and the spectral norm (i.e. the Fourier L₁-norm) of a Boolean function. This answers an open question of Cheung, Hatami, Hosseini and Shirley (CCC 2023). As immediate consequences, we obtain the following results.  
- We give an exponential separation between the logarithm of the randomized and the deterministic parity decision tree size. This is in sharp contrast with the standard binary decision tree setting where the logarithms of randomized and deterministic decision tree size are essentially polynomially related, as shown recently by Chattopadhyay, Dahiya, Mande, Radhakrishnan, and Sanyal (STOC 2023). 
- We give an exponential separation between the approximate and the exact spectral norm for Boolean functions.
- We give an exponential separation for XOR functions between the deterministic communication complexity with oracle access to Equality function (D^EQ) and randomized communication complexity. Previously, such a separation was known for general Boolean matrices by Chattopadhyay, Lovett, and Vinyals (CCC 2019) using the Integer Inner Product (IIP) function. 
- Finally, our method gives an elementary and short proof for the mentioned exponential D^EQ lower bound of Chattopadhyay, Lovett, and Vinyals for Integer Inner Product (IIP).

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Oracles and decision trees
Keywords
  • Boolean function complexity
  • parity decision trees
  • randomized communication complexity

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