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Low-Depth Arithmetic Circuit Lower Bounds: Bypassing Set-Multilinearization

Authors Prashanth Amireddy, Ankit Garg, Neeraj Kayal, Chandan Saha, Bhargav Thankey

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Author Details

Prashanth Amireddy
  • Harvard University, Cambridge, MA, USA
Ankit Garg
  • Microsoft Research, Bangalore, India
Neeraj Kayal
  • Microsoft Research, Bangalore, India
Chandan Saha
  • Indian Institute of Science, Bangalore, India
Bhargav Thankey
  • Indian Institute of Science, Bangalore, India

Funding

  • Amireddy, Prashanth: Supported in part by a Simons Investigator Award and NSF Award CCF 2152413 to Madhu Sudan. A part of this work was done while the author was a research fellow at Microsoft Research, India.
  • Saha, Chandan: Partially supported by a MATRICS grant of the Science and Engineering Research Board, DST, India.
  • Thankey, Bhargav: Supported by the Prime Minister’s Research Fellowship, India.

Acknowledgements

We would like to thank the anonymous reviewers for their valuable feedback.

Cite As Get BibTex

Prashanth Amireddy, Ankit Garg, Neeraj Kayal, Chandan Saha, and Bhargav Thankey. Low-Depth Arithmetic Circuit Lower Bounds: Bypassing Set-Multilinearization. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.12

Abstract

A recent breakthrough work of Limaye, Srinivasan and Tavenas [Nutan Limaye et al., 2021] proved superpolynomial lower bounds for low-depth arithmetic circuits via a "hardness escalation" approach: they proved lower bounds for low-depth set-multilinear circuits and then lifted the bounds to low-depth general circuits. In this work, we prove superpolynomial lower bounds for low-depth circuits by bypassing the hardness escalation, i.e., the set-multilinearization, step. As set-multilinearization comes with an exponential blow-up in circuit size, our direct proof opens up the possibility of proving an exponential lower bound for low-depth homogeneous circuits by evading a crucial bottleneck. Our bounds hold for the iterated matrix multiplication and the Nisan-Wigderson design polynomials. We also define a subclass of unrestricted depth homogeneous formulas which we call unique parse tree (UPT) formulas, and prove superpolynomial lower bounds for these. This significantly generalizes the superpolynomial lower bounds for regular formulas [Neeraj Kayal et al., 2014; Hervé Fournier et al., 2015].

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • arithmetic circuits
  • low-depth circuits
  • lower bounds
  • shifted partials

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