research-article

Approaching the Chasm at Depth Four

Authors:

Pritish Kamath,

Ramprasad SaptharishiAuthors Info & Claims

Journal of the ACM (JACM), Volume 61, Issue 6

Article No.: 33, Pages 1 - 16

https://doi.org/10.1145/2629541

Published: 17 December 2014 Publication History

Abstract

Agrawal and Vinay [2008], Koiran [2012], and Tavenas [2013] have recently shown that an exp (ω(√n log n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of × gates having fanin bounded by √n translates to a superpolynomial lower bound for general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin.

We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by √n computing the permanent (or the determinant) must be of size exp,(Ω(√n)).

References

[1]

Manindra Agrawal and V. Vinay. 2008. Arithmetic Circuits: A chasm at depth four. In Proceedings of Foundations of Computer Science (FOCS). 67--75.

Digital Library

[2]

Leandro Caniglia, Jorge A. Guccione, and Juan J. Guccione. 1990. Ideals of generic minors. Commut. Algebra 18, 2633--2640.

[3]

David A. Cox, John B. Little, and Donal O'Shea. 2007. Ideals, Varieties and Algorithms. Springer.

Digital Library

[4]

Hervé Fournier, Nutan Limaye, Guillaume Malod, and Srikanth Srinivasan. 2013. Lower bounds for depth 4 formulas computing iterated matrix multiplication. Electronic Colloquium on Computational Complex. (ECCC) 100.

[5]

Dima Grigoriev and Marek Karpinski. 1998. An exponential lower bound for depth 3 arithmetic circuits. In Proceedings of Symposium on Theory of Computing (STOC). 577--582.

Digital Library

[6]

Dima Grigoriev and Alexander A. Razborov. 2000. Exponential lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields. Appl. Algebra Eng. Commun. Comput. 10, 6, 465--487.

[7]

Ankit Gupta, Neeraj Kayal, and Youming Qiao. 2013. Random arithmetic formulas can be reconstructed efficiently. In Proceedings of Conference on Computational Complexity (CCC). 1--9.

[8]

Neeraj Kayal. 2012a. Affine projections of polynomials. In Proceedings of Symposium on Theory of Computing (STOC). 643--662.

Digital Library

[9]

Neeraj Kayal. 2012b. An exponential lower bound for the sum of powers of bounded degree polynomials. Electronic Colloquium on Computational Complexity (ECCC). 19.

[10]

Neeraj Kayal, Chandan Saha, and Ramprasad Saptharishi. 2013. A super-polynomial lower bound for regular arithmetic formulas. Electronic Colloquium on Computational Complexity (ECCC). 20.

[11]

Pascal Koiran. 2012. Arithmetic circuits: The chasm at depth four gets wider. Theoret. Comput. Sci. 448, 56--65.

Digital Library

[12]

Mrinal Kumar and Shubhangi Saraf. 2013. Lower bounds for depth 4 homogenous circuits with bounded top fanin. Electronic Colloquium on Computational Complexity (ECCC) 20, 68.

[13]

H. Narasimhan. 1986. The irreducibility of ladder determinantal varieties. J. Algebra 102 (1986), 162--185.

[14]

Noam Nisan and Avi Wigderson. 1997. Lower bounds on arithmetic circuits via partial derivatives. Computat. Complex. 6, 3, 217--234.

Digital Library

[15]

Ran Raz. 2009. Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM 56, 2.

Digital Library

[16]

Ran Raz and Amir Yehudayoff. 2008. Lower bounds and separations for constant depth mutilinear circuits. In Proceedings of the IEEE 23rd Annual Conference on Computation Complexity (CCC).

Digital Library

[17]

Dan Romik. 2000. Stirling's approximation for n&excl;: The ultimate short proof&quest; Amer. Math. Monthly 107, 6, 556--557.

[18]

Amir Shpilka and Avi Wigderson. 2001. Depth-3 arithmetic circuits over fields of characteristic zero. Computat. Complex. 10, 1, 1--27.

Digital Library

[19]

Bernd Sturmfels. 1990. Gröbner bases and Stanley decompositions of determinantal rings. Mathematische Zeitschrift 209, 137--144.

[20]

Sébastien Tavenas. 2013. Improved bounds for reduction to depth 4 and depth 3. In Proceedings of Mathematical Foundations of Computer Science (MFCS). 813--824.

[21]

Leslie G. Valiant. 1979. Completeness classes in algebra. In Proceedings of Symposium on Theory of Computing (STOC). 249--261.

Digital Library

Cited By

Bhargav CDutta SSaxena N(2024)Improved Lower Bound, and Proof Barrier, for Constant Depth Algebraic CircuitsACM Transactions on Computation Theory10.1145/368995716:4(1-22)Online publication date: 11-Nov-2024
https://doi.org/10.1145/3689957
Fournier HLimaye NSrinivasan STavenas SMohar BShinkar IO'Donnell R(2024)On the Power of Homogeneous Algebraic FormulasProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649760(141-151)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649760
Limaye NSrinivasan STavenas S(2024)Superpolynomial Lower Bounds Against Low-Depth Algebraic CircuitsCommunications of the ACM10.1145/361109467:2(101-108)Online publication date: 25-Jan-2024
https://dl.acm.org/doi/10.1145/3611094
Show More Cited By

Index Terms

Approaching the Chasm at Depth Four
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Representation of mathematical objects
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on polynomials

Recommendations

A super-polynomial lower bound for regular arithmetic formulas
STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

We consider arithmetic formulas consisting of alternating layers of addition (+) and multiplication (×) gates such that the fanin of all the gates in any fixed layer is the same. Such a formula Φ which additionally has the property that its formal/...
Marginal hitting sets imply super-polynomial lower bounds for permanent
ITCS '12: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference

Suppose f is a univariate polynomial of degree r = r(n) that is computed by a size n arithmetic circuit. It is a basic fact of algebra that a nonzero univariate polynomial of degree r can vanish on at most r points. This implies that for checking ...
Hardness-randomness tradeoffs for bounded depth arithmetic circuits
STOC '08: Proceedings of the fortieth annual ACM symposium on Theory of computing

In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x₁,...,x_m) that cannot be ...

Comments

Information & Contributors

Information

Published In

cover image Journal of the ACM

Journal of the ACM Volume 61, Issue 6

November 2014

285 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/2700084

Editor:
Victor Vianu
University of California, San Diego

Issue’s Table of Contents

Copyright © 2014 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 17 December 2014

Accepted: 01 January 2014

Revised: 01 January 2014

Received: 01 September 2013

Published in JACM Volume 61, Issue 6

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article
Research
Refereed

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

41
Total Citations
View Citations
496
Total Downloads

Downloads (Last 12 months)32
Downloads (Last 6 weeks)2

Reflects downloads up to 10 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Bhargav CDutta SSaxena N(2024)Improved Lower Bound, and Proof Barrier, for Constant Depth Algebraic CircuitsACM Transactions on Computation Theory10.1145/368995716:4(1-22)Online publication date: 11-Nov-2024
https://doi.org/10.1145/3689957
Fournier HLimaye NSrinivasan STavenas SMohar BShinkar IO'Donnell R(2024)On the Power of Homogeneous Algebraic FormulasProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649760(141-151)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649760
Limaye NSrinivasan STavenas S(2024)Superpolynomial Lower Bounds Against Low-Depth Algebraic CircuitsCommunications of the ACM10.1145/361109467:2(101-108)Online publication date: 25-Jan-2024
https://dl.acm.org/doi/10.1145/3611094
Limaye NSrinivasan STavenas S(2022)Guest ColumnACM SIGACT News10.1145/3544979.354498953:2(40-62)Online publication date: 25-Jul-2022
https://dl.acm.org/doi/10.1145/3544979.3544989
Tavenas SLimaye NSrinivasan SLeonardi SGupta A(2022)Set-multilinear and non-commutative formula lower bounds for iterated matrix multiplicationProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520044(416-425)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520044
Limaye NSrinivasan STavenas S(2022)Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00083(804-814)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00083
Origlia ARossi SMartino SCutugno FChiacchio M(2021)Multiple-source Data Collection and Processing into a Graph Database Supporting Cultural Heritage ApplicationsJournal on Computing and Cultural Heritage 10.1145/346574114:4(1-27)Online publication date: 16-Jul-2021
https://dl.acm.org/doi/10.1145/3465741
Ngo VDuong TNguyen TNguyen PConlan O(2021)An Efficient Classification Algorithm for Traditional Textile Patterns from Different Cultures Based on StructuresJournal on Computing and Cultural Heritage 10.1145/346538114:4(1-22)Online publication date: 16-Jul-2021
https://dl.acm.org/doi/10.1145/3465381
Sprugnoli RGuerini MMoretti GTonelli S(2021)Are these Artworks Similar? Analysing Visitors’ Judgements on Aesthetic Perception with a Digital GameJournal on Computing and Cultural Heritage 10.1145/346166314:4(1-14)Online publication date: 16-Jul-2021
https://dl.acm.org/doi/10.1145/3461663
Fafalios PPetrakis KSamaritakis GDoerr KKritsotaki ATzitzikas YDoerr M(2021)FAST CATJournal on Computing and Cultural Heritage 10.1145/346146014:4(1-20)Online publication date: 16-Jul-2021
https://dl.acm.org/doi/10.1145/3461460
Show More Cited By

View Options

Get Access

Login options

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

Full Access

Get this Article

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Issue’s Table of Contents