skip to main content
research-article

On the Power of Border of Depth-3 Arithmetic Circuits

Published:11 February 2020Publication History
Skip Abstract Section

Abstract

We show that over the field of complex numbers, every homogeneous polynomial of degree d can be approximated (in the border complexity sense) by a depth-3 arithmetic circuit of top fan-in at most 2. This is quite surprising, since there exist homogeneous polynomials P on n variables of degree 2, such that any depth-3 arithmetic circuit computing P must have top fan-in at least Ω (n).

As an application, we get a new tradeoff between the top fan-in and formal degree in an approximate analog of the celebrated depth reduction result of Gupta, Kamath, Kayal, and Saptharishi [7, 10]. Formally, we show that if a degree d homogeneous polynomial P can be computed by an arithmetic circuit of size sd, then for every td, P is in the border of a depth-3 circuit of top fan-in sO(t) and formal degree sO(d/t). To the best of our knowledge, the upper bound on the top fan-in in the original proof of Reference [7] is always at least sΩ (√d), regardless of the formal degree.

References

  1. Manindra Agrawal and V. Vinay. 2008. Arithmetic circuits: A chasm at depth four. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS’08). 67--75. DOI:https://doi.org/10.1109/FOCS.2008.32Google ScholarGoogle Scholar
  2. Eric Allender and Fengming Wang. 2016. On the power of algebraic branching programs of width two. Comput. Complex. 25, 1 (2016), 217--253. DOI:https://doi.org/10.1007/s00037-015-0114-7Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Michael Ben-Or and Richard Cleve. 1988. Computing algebraic formulas using a constant number of registers. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC’88). 254--257.Google ScholarGoogle Scholar
  4. Peter Bürgisser. 2004. The complexity of factors of multivariate polynomials. Found. Comput. Math. 4, 4 (2004), 369--396. DOI:https://doi.org/10.1007/s10208-002-0059-5Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Mahdi Cheraghchi, Elena Grigorescu, Brendan Juba, Karl Wimmer, and Ning Xie. 2018. AC0 ○ MOD2 lower bounds for the Boolean inner product. J. Comput. Syst. Sci. 97 (2018), 45--59. DOI:https://doi.org/10.1016/j.jcss.2018.04.006Google ScholarGoogle ScholarCross RefCross Ref
  6. Ismor Fischer. 1994. Sums of like powers of multivariate linear forms. Math. Mag. 67, 1 (1994), 59--61. http://www.jstor.org/stable/2690560Google ScholarGoogle ScholarCross RefCross Ref
  7. Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. 2016. Arithmetic circuits: A chasm at depth 3. SIAM J. Comput. 45, 3 (2016), 1064--1079. DOI:https://doi.org/10.1137/140957123Google ScholarGoogle ScholarCross RefCross Ref
  8. Noam Nisan and Avi Wigderson. 1997. Lower bounds on arithmetic circuits via partial derivatives. Comput. Complex. 6, 3 (1997), 217--234. DOI:https://doi.org/10.1007/BF01294256 Available on citeseer:10.1.1.90.2644.Google ScholarGoogle ScholarDigital LibraryDigital Library
  9. Amir Shpilka. 2002. Affine projections of symmetric polynomials. J. Comput. System Sci. 65, 4 (2002), 639--659. DOI:https://doi.org/10.1016/S0022-0000(02)00021-1 Special Issue on Complexity 2001.Google ScholarGoogle ScholarDigital LibraryDigital Library
  10. Sébastien Tavenas. 2015. Improved bounds for reduction to depth 4 and depth 3. Inf. Comput. 240 (2015), 2--11. DOI:https://doi.org/10.1016/j.ic.2014.09.004Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Karl Bringmann, Christian Ikenmeyer, and Jeroen Zuiddam. 2018. On algebraic branching programs of small width. J. ACM 65, 5 (2018), 32:1--32:29. https://doi.org/10.1145/3209663Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. On the Power of Border of Depth-3 Arithmetic Circuits

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in

      Full Access

      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 12, Issue 1
        March 2020
        199 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3376904
        Issue’s Table of Contents

        Copyright © 2020 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 11 February 2020
        • Accepted: 1 September 2019
        • Revised: 1 July 2019
        • Received: 1 May 2018
        Published in toct Volume 12, Issue 1

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • research-article
        • Research
        • Refereed

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader

      HTML Format

      View this article in HTML Format .

      View HTML Format
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!