On the Partial Derivative Method Applied to Lopsided Set-Multilinear Polynomials

Authors Nutan Limaye, Srikanth Srinivasan, Sébastien Tavenas



PDF
Thumbnail PDF

File

LIPIcs.CCC.2022.32.pdf
  • Filesize: 0.75 MB
  • 23 pages

Document Identifiers

Author Details

Nutan Limaye
  • Computer Science Department, IT University of Copenhagen, Denmark
Srikanth Srinivasan
  • Department of Computer Science, Aarhus University, Denmark
  • On leave from Department of Mathematics, IIT Bombay, India
Sébastien Tavenas
  • Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, France

Acknowledgements

We would like to thank the anonymous reviewers of the paper for their comments which helped us improve the presentation in the paper. We would also like to thank Niranjan Balachandran for his comments and an alternate proof of a combinatorial lemma in the paper, and Swastik Kopparty and Shachar Lovett for discussions.

Cite AsGet BibTex

Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. On the Partial Derivative Method Applied to Lopsided Set-Multilinear Polynomials. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 32:1-32:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CCC.2022.32

Abstract

We make progress on understanding a lower bound technique that was recently used by the authors to prove the first superpolynomial constant-depth circuit lower bounds against algebraic circuits. More specifically, our previous work applied the well-known partial derivative method in a new setting, that of lopsided set-multilinear polynomials. A set-multilinear polynomial P ∈ 𝔽[X_1,…,X_d] (for disjoint sets of variables X_1,…,X_d) is a linear combination of monomials, each of which contains one variable from X_1,…,X_d. A lopsided space of set-multilinear polynomials is one where the sets X_1,…,X_d are allowed to have different sizes (we use the adjective "lopsided" to stress this feature). By choosing a suitable lopsided space of polynomials, and using a suitable version of the partial-derivative method for proving lower bounds, we were able to prove constant-depth superpolynomial set-multilinear formula lower bounds even for very low-degree polynomials (as long as d is a growing function of the number of variables N). This in turn implied lower bounds against general formulas of constant-depth. A priori, there is nothing stopping these techniques from giving us lower bounds against algebraic formulas of any depth. We investigate the extent to which this lower bound can extend to greater depths. We prove the following results. 1) We observe that our choice of the lopsided space and the kind of partial-derivative method used can be modeled as the choice of a multiset W ⊆ [-1,1] of size d. Our first result completely characterizes, for any product-depth Δ, the best lower bound we can prove for set-multilinear formulas of product-depth Δ in terms of some combinatorial properties of W, that we call the depth-Δ tree bias of W. 2) We show that the maximum depth-3 tree bias, over multisets W of size d, is Θ(d^{1/4}). This shows a stronger formula lower bound of N^{Ω(d^{1/4})} for set-multilinear formulas of product-depth 3, and also puts a non-trivial constraint on the best lower bounds we can hope to prove at this depth in this framework (a priori, we could have hoped to prove a lower bound of N^{Ω(Δ d^{1/Δ})} at product-depth Δ). 3) Finally, we show that for small Δ, our proof technique cannot hope to prove lower bounds of the form N^{Ω(d^{1/poly(Δ)})}. This seems to strongly hint that new ideas will be required to prove lower bounds for formulas of unbounded depth.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Partial Derivative Method
  • Barriers to Lower Bounds

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Miklós Ajtai. Σ₁¹-formulae on finite structures. Annals of Pure and Applied Logic, 24(1):1-48, 1983. URL: https://doi.org/10.1016/0168-0072(83)90038-6.
  2. Peter Bürgisser. Cook’s versus valiant’s hypothesis. Theoretical Computer Science, 235(1):71-88, 2000. Google Scholar
  3. Peter Bürgisser, Michael Clausen, and Mohammad Amin Shokrollahi. Algebraic complexity theory, volume 315 of Grundlehren der mathematischen Wissenschaften. Springer, 1997. Google Scholar
  4. Klim Efremenko, Ankit Garg, Rafael Mendes de Oliveira, and Avi Wigderson. Barriers for rank methods in arithmetic complexity. In Proceedings of the 9th Innovations in Theoretical Computer Science Conference (ITCS), volume 94 of LIPIcs, pages 1-19, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.1.
  5. Klim Efremenko, J. M. Landsberg, Hal Schenck, and Jerzy Weyman. The method of shifted partial derivatives cannot separate the permanent from the determinant. Mathematics of Computation, 87(312):2037-2045, 2018. URL: https://doi.org/10.1090/mcom/3284.
  6. Michael A. Forbes, Amir Shpilka, and Ben Lee Volk. Succinct hitting sets and barriers to proving lower bounds for algebraic circuits. Theory of Computing, 14(1):1-45, 2018. URL: https://doi.org/10.4086/toc.2018.v014a018.
  7. Merrick Furst, James B. Saxe, and Michael Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical systems theory, 17(1):13-27, 1984. URL: https://doi.org/10.1007/BF01744431.
  8. Dima Grigoriev and Marek Karpinski. An exponential lower bound for depth 3 arithmetic circuits. In Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing (STOC), pages 577-582, 1998. URL: https://doi.org/10.1145/276698.276872.
  9. Dima Grigoriev and Alexander A. Razborov. Exponential lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields. Applicable Algebra in Engineering, Communication and Computing, 10(6):465-487, 2000. URL: https://doi.org/10.1007/s002009900021.
  10. Joshua A. Grochow. Unifying known lower bounds via geometric complexity theory. Computational Complexity, 24(2):393-475, 2015. URL: https://doi.org/10.1007/s00037-015-0103-x.
  11. Joshua A. Grochow, Mrinal Kumar, Michael E. Saks, and Shubhangi Saraf. Towards an algebraic natural proofs barrier via polynomial identity testing. CoRR, abs/1701.01717, 2017. URL: http://arxiv.org/abs/1701.01717.
  12. Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Arithmetic circuits: A chasm at depth 3. SIAM Journal of Computing, 45(3):1064-1079, 2016. URL: https://doi.org/10.1137/140957123.
  13. Johan Håstad. Almost optimal lower bounds for small depth circuits. In Proceedings of the 18th Annual ACM Symposium on Theory of Computing (STOC), pages 6-20, 1986. URL: https://doi.org/10.1145/12130.12132.
  14. Stasys Jukna. Extremal Combinatorics - With Applications in Computer Science. Texts in Theoretical Computer Science. An EATCS Series. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-17364-6.
  15. Guillaume Lagarde, Nutan Limaye, and Srikanth Srinivasan. Lower bounds and PIT for non-commutative arithmetic circuits with restricted parse trees. Computational Complexity, 28(3):471-542, 2019. URL: https://doi.org/10.1007/s00037-018-0171-9.
  16. Guillaume Lagarde, Guillaume Malod, and Sylvain Perifel. Non-commutative computations: lower bounds and polynomial identity testing. Chicago Journal of Theoretical Computer Science, 2019:2, 2019. URL: http://cjtcs.cs.uchicago.edu/articles/2019/2/contents.html.
  17. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. Set-multilinear and non-commutative formula lower bounds for iterated matrix multiplication. To appear in STOC 2022. Electron. Colloquium Comput. Complex., page 94, 2021. URL: https://eccc.weizmann.ac.il/report/2021/094.
  18. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. Superpolynomial lower bounds against low-depth algebraic circuits. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 804-814. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00083.
  19. Nutan Limaye, Srikanth Srinivasan, and Sébastien Tavenas. Superpolynomial lower bounds against low-depth algebraic circuits. Electron. Colloquium Comput. Complex., page 81, 2021. URL: https://eccc.weizmann.ac.il/report/2021/081.
  20. Noam Nisan and Avi Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Computational Complexity, 6(3):217-234, 1997. URL: https://doi.org/10.1007/BF01294256.
  21. Ran Raz. Multi-linear formulas for permanent and determinant are of super-polynomial size. Journal of the ACM, 56(2):8:1-8:17, 2009. URL: https://doi.org/10.1145/1502793.1502797.
  22. Ran Raz. Tensor-rank and lower bounds for arithmetic formulas. Journal of the ACM, 60(6):40:1-40:15, 2013. URL: https://doi.org/10.1145/2535928.
  23. Alexander A. Razborov. Lower bounds on the size of constant-depth networks over a complete basis with logical addition. Mathematicheskie Zametki, 41(2):598-607, 1986. Google Scholar
  24. Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity. Github survey, 2015. URL: https://github.com/dasarpmar/lowerbounds-survey/releases/.
  25. Andrew Drucker Scott Aaronson. Arithmetic natural proofs theory is sought. Blog post, 24(1):1-48, 2008. Google Scholar
  26. Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5:207-388, 2010. URL: https://doi.org/10.1561/0400000039.
  27. Roman Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC), pages 77-82, 1987. URL: https://doi.org/10.1145/28395.28404.
  28. Leslie G. Valiant. Completeness classes in algebra. In Proceedings of the 11h Annual ACM Symposium on Theory of Computing (STOC), pages 249-261. ACM, 1979. URL: https://doi.org/10.1145/800135.804419.