research-article

Lower bounds for depth 4 formulas computing iterated matrix multiplication

Authors:

Hervé Fournier,

Guillaume Malod,

Srikanth SrinivasanAuthors Info & Claims

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

Pages 128 - 135

https://doi.org/10.1145/2591796.2591824

Published: 31 May 2014 Publication History

Abstract

We study the arithmetic complexity of iterated matrix multiplication. We show that any multilinear homogeneous depth 4 arithmetic formula computing the product of d generic matrices of size n × n, IMM_n,d, has size n^Ω(√d) as long as d = n^O(1). This improves the result of Nisan and Wigderson (Computational Complexity, 1997) for depth 4 set-multilinear formulas.

We also study ΣΠ^[O(d/t)] ΣΠ^[t] formulas, which are depth 4 formulas with the stated bounds on the fan-ins of the Π gates. A recent depth reduction result of Tavenas (MFCS, 2013) shows that any n-variate degree d = n^O(1) polynomial computable by a circuit of size poly(n) can also be computed by a depth 4 ΣΠ^[O(d/t)] ΣΠ^[t] formula of top fan-in n^O(d/t). We show that any such formula computing IMM_n,d has top fan-in n^Ω(d/t), proving the optimality of Tavenas' result. This also strengthens a result of Kayal, Saha, and Saptharishi (ECCC, 2013) which gives a similar lower bound for an explicit polynomial in VNP.

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References

[1]

M. Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In FOCS, pages 67--75, 2008.

Digital Library

[2]

S. Chillara and P. Mukhopadhyay. Depth-4 lower bounds, determinantal complexity: A unified approach. In STACS, pages 239--250, 2014.

[3]

Z. Dvir, G. Malod, S. Perifel, and A. Yehudayoff. Separating multilinear branching programs and formulas. In H. J. Karloff and T. Pitassi, editors, STOC, pages 615--624. ACM, 2012.

Digital Library

[4]

M. L. Furst, J. B. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17(1):13--27, 1984.

[5]

A. Gupta, P. Kamath, N. Kayal, and R. Saptharishi. Approaching the chasm at depth four. In Proceedings of the Conference on Computational Complexity (CCC), 2013.

[6]

V. Guruswami. Introduction to coding theory, Lecture 2: Gilbert-Varshamov bound. http://www.cs.cmu.edu/venkatg/teaching/codingtheory/, 2010.

[7]

J. Håstad. Computational limitations of small-depth circuits. The MIT Press, Cambridge(MA)-London, 1987.

Digital Library

[8]

N. Kayal. An exponential lower bound for the sum of powers of bounded degree polynomials. Electronic Colloquium on Computational Complexity (ECCC), 19:81, 2012.

[9]

N. Kayal, N. Limaye, C. Saha, and S. Srinivasan. Super-polynomial lower bounds for depth-4 homogeneous arithmetic formulas. Manuscript, 2013.

[10]

N. Kayal, C. Saha, and R. Saptharishi. A super-polynomial lower bound for regular arithmetic formulas. Electronic Colloquium on Computational Complexity (ECCC), 20:91, 2013.

[11]

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

Digital Library

[12]

M. Kumar and S. Saraf. Lower Bounds for Depth 4 Homogenous Circuits with Bounded Top Fanin. Electronic Colloquium on Computational Complexity (ECCC), 20:68, 2013.

[13]

M. Kumar and S. Saraf. Superpolynomial lower bounds for general homogeneous depth 4 arithmetic circuits. CoRR, abs/1312.5978, 2013.

[14]

M. Kumar and S. Saraf. The Limits of Depth Reduction for Arithmetic Formulas: It's all about the top fan-in. Electronic Colloquium on Computational Complexity (ECCC), 2013.

[15]

M. Mahajan and V. Vinay. Determinant: Combinatorics, algorithms, and complexity. Chicago J. Theor. Comput. Sci., 1997, 1997.

[16]

G. Malod and N. Portier. Characterizing valiant's algebraic complexity classes. J. Complex., 24(1):16--38, 2008.

Digital Library

[17]

N. Nisan and A. Wigderson. Lower bounds on arithmetic circuits via partial derivatives. Computational Complexity, 6(3):217--234, 1997.

Digital Library

[18]

R. Raz. Separation of multilinear circuit and formula size. Theory of Computing, 2(1):121--135, 2006.

[19]

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

Digital Library

[20]

R. Raz. Tensor-rank and lower bounds for arithmetic formulas. In L. J. Schulman, editor, STOC, pages 659--666. ACM, 2010.

Digital Library

[21]

R. Raz and A. Yehudayoff. Balancing syntactically multilinear arithmetic circuits. Computational Complexity, 17(4):515--535, 2008.

Digital Library

[22]

R. Raz and A. Yehudayoff. Lower bounds and separations for constant depth multilinear circuits. Computational Complexity, 18(2):171--207, 2009.

[23]

A. Shpilka and A. Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3-4):207--388, 2010.

Digital Library

[24]

S. Tavenas. Improved bounds for reduction to depth 4 and 3. In Mathematical Foundations of Computer Science (MFCS), 2013.

[25]

S. Toda. Classes of Arithmetic Circuits Capturing the Complexity of Computing the Determinant. IEICE Transactions on Information and Systems, E75-D:116--124, 1992.

[26]

L. G. Valiant. Completeness Classes in Algebra. In STOC '79: Proceedings of the eleventh annual ACM symposium on Theory of computing, pages 249--261, New York, NY, USA, 1979. ACM Press.

Digital Library

[27]

L. G. Valiant. Reducibility by Algebraic Projections. In Logic and Algorithmic: an International Symposium held in honor of Ernst Specker, volume 30 of Monographies de l'Enseignement Mathémathique, pages 365--380, 1982.

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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
Fijalkow NLagarde GOhlmann PSerre O(2021)Lower Bounds for Arithmetic Circuits via the Hankel Matrixcomputational complexity10.1007/s00037-021-00214-130:2Online publication date: 8-Oct-2021
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Kayal NNair VSaha C(2020)Separation Between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth-three CircuitsACM Transactions on Computation Theory10.1145/336992812:1(1-27)Online publication date: 11-Feb-2020
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Lower bounds for depth 4 formulas computing iterated matrix multiplication

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cover image ACM Conferences

STOC '14: Proceedings of the forty-sixth annual ACM symposium on Theory of computing

May 2014

984 pages

ISBN:9781450327107

DOI:10.1145/2591796

Program Chair:
David Shmoys
Cornell University

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]

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Published: 31 May 2014

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Indo-French Centre for the Promotion of Advanced Research

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May 31 - June 3, 2014

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Cited By

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
Fijalkow NLagarde GOhlmann PSerre O(2021)Lower Bounds for Arithmetic Circuits via the Hankel Matrixcomputational complexity10.1007/s00037-021-00214-130:2Online publication date: 8-Oct-2021
https://doi.org/10.1007/s00037-021-00214-1
Kayal NNair VSaha C(2020)Separation Between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth-three CircuitsACM Transactions on Computation Theory10.1145/336992812:1(1-27)Online publication date: 11-Feb-2020
https://dl.acm.org/doi/10.1145/3369928
Garg AKayal NSaha C(2020)Learning sums of powers of low-degree polynomials in the non-degenerate case2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS46700.2020.00087(889-899)Online publication date: Nov-2020
https://doi.org/10.1109/FOCS46700.2020.00087
Chillara SMukhopadhyay P(2019)Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approachcomputational complexity10.1007/s00037-019-00185-4Online publication date: 28-May-2019
https://doi.org/10.1007/s00037-019-00185-4
Ghosal PRaghavendra Rao B(2019)On Proving Parameterized Size Lower Bounds for Multilinear Algebraic ModelsComputing and Combinatorics10.1007/978-3-030-26176-4_15(178-192)Online publication date: 21-Jul-2019
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Chou CKumar MSolomon NServedio R(2018)Hardness vs randomness for bounded depth arithmetic circuitsProceedings of the 33rd Computational Complexity Conference10.5555/3235586.3235599(1-17)Online publication date: 22-Jun-2018
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Alon NKumar MVolk BServedio R(2018)Unbalancing sets and an almost quadratic lower bound for syntactically multilinear arithmetic circuitsProceedings of the 33rd Computational Complexity Conference10.5555/3235586.3235597(1-16)Online publication date: 22-Jun-2018
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Anderson MForbes MSaptharishi RShpilka AVolk B(2018)Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching ProgramsACM Transactions on Computation Theory10.1145/317070910:1(1-30)Online publication date: 10-Jan-2018
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Kumar MSaptharishi R(2017)An exponential lower bound for homogeneous depth-5 circuits over finite fieldsProceedings of the 32nd Computational Complexity Conference10.5555/3135595.3135626(1-30)Online publication date: 9-Jul-2017
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