8 Search Results for "Fournier, Hervé"


Document
Towards Optimal Depth-Reductions for Algebraic Formulas

Authors: Hervé Fournier, Nutan Limaye, Guillaume Malod, Srikanth Srinivasan, and Sébastien Tavenas

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)


Abstract
Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula. Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth O(log d) and size poly(s). Our proof shows that this result also holds in the highly restricted setting of monotone, non-commutative algebraic formulas. This improves on previous results in the regime when d is small (i.e., d = s^o(1)). In particular, for the setting of d = O(log s), along with a result of Raz (STOC 2010, JACM 2013), our result implies the same depth reduction even for inhomogeneous formulas. This is particularly interesting in light of recent algebraic formula lower bounds, which work precisely in this "low-degree" and "low-depth" setting. We also show that these results cannot be improved in the monotone setting, even for commutative formulas.

Cite as

Hervé Fournier, Nutan Limaye, Guillaume Malod, Srikanth Srinivasan, and Sébastien Tavenas. Towards Optimal Depth-Reductions for Algebraic Formulas. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 28:1-28:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{fournier_et_al:LIPIcs.CCC.2023.28,
  author =	{Fournier, Herv\'{e} and Limaye, Nutan and Malod, Guillaume and Srinivasan, Srikanth and Tavenas, S\'{e}bastien},
  title =	{{Towards Optimal Depth-Reductions for Algebraic Formulas}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{28:1--28:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.28},
  URN =		{urn:nbn:de:0030-drops-182986},
  doi =		{10.4230/LIPIcs.CCC.2023.28},
  annote =	{Keywords: Algebraic formulas, depth-reduction}
}
Document
Track A: Algorithms, Complexity and Games
Low-Depth Arithmetic Circuit Lower Bounds: Bypassing Set-Multilinearization

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

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)


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].

Cite as

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)


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@InProceedings{amireddy_et_al:LIPIcs.ICALP.2023.12,
  author =	{Amireddy, Prashanth and Garg, Ankit and Kayal, Neeraj and Saha, Chandan and Thankey, Bhargav},
  title =	{{Low-Depth Arithmetic Circuit Lower Bounds: Bypassing Set-Multilinearization}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{12:1--12:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.12},
  URN =		{urn:nbn:de:0030-drops-180642},
  doi =		{10.4230/LIPIcs.ICALP.2023.12},
  annote =	{Keywords: arithmetic circuits, low-depth circuits, lower bounds, shifted partials}
}
Document
Polynomial Identity Testing via Evaluation of Rational Functions

Authors: Dieter van Melkebeek and Andrew Morgan

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)


Abstract
We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. In spite of the univariate nature, we establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials in the abscissas. We study the power of the generator by characterizing its vanishing ideal, i.e., the set of polynomials that it fails to hit. Capitalizing on the univariate nature, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition size of set-multi-linearity in the vanishing ideal. Inspired by an alternating algebra representation, we develop a structured deterministic membership test for the vanishing ideal. As a proof of concept we rederive known derandomization results based on the generator by Shpilka and Volkovich, and present a new application for read-once oblivious arithmetic branching programs that provably transcends the usual combinatorial techniques.

Cite as

Dieter van Melkebeek and Andrew Morgan. Polynomial Identity Testing via Evaluation of Rational Functions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 119:1-119:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{vanmelkebeek_et_al:LIPIcs.ITCS.2022.119,
  author =	{van Melkebeek, Dieter and Morgan, Andrew},
  title =	{{Polynomial Identity Testing via Evaluation of Rational Functions}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{119:1--119:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.119},
  URN =		{urn:nbn:de:0030-drops-157158},
  doi =		{10.4230/LIPIcs.ITCS.2022.119},
  annote =	{Keywords: Derandomization, Gr\"{o}bner Basis, Lower Bounds, Polynomial Identity Testing}
}
Document
Nonnegative Rank Measures and Monotone Algebraic Branching Programs

Authors: Hervé Fournier, Guillaume Malod, Maud Szusterman, and Sébastien Tavenas

Published in: LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)


Abstract
Inspired by Nisan’s characterization of noncommutative complexity (Nisan 1991), we study different notions of nonnegative rank, associated complexity measures and their link with monotone computations. In particular we answer negatively an open question of Nisan asking whether nonnegative rank characterizes monotone noncommutative complexity for algebraic branching programs. We also prove a rather tight lower bound for the computation of elementary symmetric polynomials by algebraic branching programs in the monotone setting or, equivalently, in the homogeneous syntactically multilinear setting.

Cite as

Hervé Fournier, Guillaume Malod, Maud Szusterman, and Sébastien Tavenas. Nonnegative Rank Measures and Monotone Algebraic Branching Programs. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 15:1-15:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{fournier_et_al:LIPIcs.FSTTCS.2019.15,
  author =	{Fournier, Herv\'{e} and Malod, Guillaume and Szusterman, Maud and Tavenas, S\'{e}bastien},
  title =	{{Nonnegative Rank Measures and Monotone Algebraic Branching Programs}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{15:1--15:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Chattopadhyay, Arkadev and Gastin, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.15},
  URN =		{urn:nbn:de:0030-drops-115774},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.15},
  annote =	{Keywords: Elementary symmetric polynomials, lower bounds}
}
Document
On the Symmetries of and Equivalence Test for Design Polynomials

Authors: Nikhil Gupta and Chandan Saha

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
In a Nisan-Wigderson design polynomial (in short, a design polynomial), every pair of monomials share a few common variables. A useful example of such a polynomial, introduced in [Neeraj Kayal et al., 2014], is the following: NW_{d,k}({x}) = sum_{h in F_d[z], deg(h) <= k}{ prod_{i=0}^{d-1}{x_{i, h(i)}}}, where d is a prime, F_d is the finite field with d elements, and k << d. The degree of the gcd of every pair of monomials in NW_{d,k} is at most k. For concreteness, we fix k = ceil[sqrt{d}]. The family of polynomials NW := {NW_{d,k} : d is a prime} and close variants of it have been used as hard explicit polynomial families in several recent arithmetic circuit lower bound proofs. But, unlike the permanent, very little is known about the various structural and algorithmic/complexity aspects of NW beyond the fact that NW in VNP. Is NW_{d,k} characterized by its symmetries? Is it circuit-testable, i.e., given a circuit C can we check efficiently if C computes NW_{d,k}? What is the complexity of equivalence test for NW, i.e., given black-box access to a f in F[{x}], can we check efficiently if there exists an invertible linear transformation A such that f = NW_{d,k}(A * {x})? Characterization of polynomials by their symmetries plays a central role in the geometric complexity theory program. Here, we answer the first two questions and partially answer the third. We show that NW_{d,k} is characterized by its group of symmetries over C, but not over R. We also show that NW_{d,k} is characterized by circuit identities which implies that NW_{d,k} is circuit-testable in randomized polynomial time. As another application of this characterization, we obtain the "flip theorem" for NW. We give an efficient equivalence test for NW in the case where the transformation A is a block-diagonal permutation-scaling matrix. The design of this algorithm is facilitated by an almost complete understanding of the group of symmetries of NW_{d,k}: We show that if A is in the group of symmetries of NW_{d,k} then A = D * P, where D and P are diagonal and permutation matrices respectively. This is proved by completely characterizing the Lie algebra of NW_{d,k}, and using an interplay between the Hessian of NW_{d,k} and the evaluation dimension.

Cite as

Nikhil Gupta and Chandan Saha. On the Symmetries of and Equivalence Test for Design Polynomials. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{gupta_et_al:LIPIcs.MFCS.2019.53,
  author =	{Gupta, Nikhil and Saha, Chandan},
  title =	{{On the Symmetries of and Equivalence Test for Design Polynomials}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{53:1--53:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.53},
  URN =		{urn:nbn:de:0030-drops-109979},
  doi =		{10.4230/LIPIcs.MFCS.2019.53},
  annote =	{Keywords: Nisan-Wigderson design polynomial, characterization by symmetries, Lie algebra, group of symmetries, circuit testability, flip theorem, equivalence test}
}
Document
Lower Bounds for Depth Three Arithmetic Circuits with Small Bottom Fanin

Authors: Neeraj Kayal and Chandan Saha

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)


Abstract
Shpilka and Wigderson (CCC 99) had posed the problem of proving exponential lower bounds for (nonhomogeneous) depth three arithmetic circuits with bounded bottom fanin over a field F of characteristic zero. We resolve this problem by proving a N^(Omega(d/t)) lower bound for (nonhomogeneous) depth three arithmetic circuits with bottom fanin at most t computing an explicit N-variate polynomial of degree d over F. Meanwhile, Nisan and Wigderson (CC 97) had posed the problem of proving superpolynomial lower bounds for homogeneous depth five arithmetic circuits. Over fields of characteristic zero, we show a lower bound of N^(Omega(sqrt(d))) for homogeneous depth five circuits (resp. also for depth three circuits) with bottom fanin at most N^(u), for any fixed u < 1. This resolves the problem posed by Nisan and Wigderson only partially because of the added restriction on the bottom fanin (a general homogeneous depth five circuit has bottom fanin at most N).

Cite as

Neeraj Kayal and Chandan Saha. Lower Bounds for Depth Three Arithmetic Circuits with Small Bottom Fanin. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 158-182, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{kayal_et_al:LIPIcs.CCC.2015.158,
  author =	{Kayal, Neeraj and Saha, Chandan},
  title =	{{Lower Bounds for Depth Three Arithmetic Circuits with Small Bottom Fanin}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{158--182},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.158},
  URN =		{urn:nbn:de:0030-drops-50617},
  doi =		{10.4230/LIPIcs.CCC.2015.158},
  annote =	{Keywords: arithmetic circuits, depth three circuits, lower bound, bottom fanin}
}
Document
A Depth-Five Lower Bound for Iterated Matrix Multiplication

Authors: Suman K. Bera and Amit Chakrabarti

Published in: LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)


Abstract
We prove that certain instances of the iterated matrix multiplication (IMM) family of polynomials with N variables and degree n require N^(Omega(sqrt(n))) gates when expressed as a homogeneous depth-five Sigma Pi Sigma Pi Sigma arithmetic circuit with the bottom fan-in bounded by N^(1/2-epsilon). By a depth-reduction result of Tavenas, this size lower bound is optimal and can be achieved by the weaker class of homogeneous depth-four Sigma Pi Sigma Pi circuits. Our result extends a recent result of Kumar and Saraf, who gave the same N^(Omega(sqrt(n))) lower bound for homogeneous depth-four Sigma Pi Sigma Pi circuits computing IMM. It is analogous to a recent result of Kayal and Saha, who gave the same lower bound for homogeneous Sigma Pi Sigma Pi Sigma circuits (over characteristic zero) with bottom fan-in at most N^(1-epsilon), for the harder problem of computing certain polynomials defined by Nisan-Wigderson designs.

Cite as

Suman K. Bera and Amit Chakrabarti. A Depth-Five Lower Bound for Iterated Matrix Multiplication. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 183-197, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{bera_et_al:LIPIcs.CCC.2015.183,
  author =	{Bera, Suman K. and Chakrabarti, Amit},
  title =	{{A Depth-Five Lower Bound for Iterated Matrix Multiplication}},
  booktitle =	{30th Conference on Computational Complexity (CCC 2015)},
  pages =	{183--197},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-81-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{33},
  editor =	{Zuckerman, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.183},
  URN =		{urn:nbn:de:0030-drops-50622},
  doi =		{10.4230/LIPIcs.CCC.2015.183},
  annote =	{Keywords: arithmetic circuits, iterated matrix multiplication, depth five circuits, lower bound}
}
Document
Monomials in arithmetic circuits: Complete problems in the counting hierarchy

Authors: Hervé Fournier, Guillaume Malod, and Stefan Mengel

Published in: LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)


Abstract
We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials. We show that these problems are complete for subclasses of the counting hierarchy which had few or no known natural complete problems before. We also study these questions for circuits computing multilinear polynomials.

Cite as

Hervé Fournier, Guillaume Malod, and Stefan Mengel. Monomials in arithmetic circuits: Complete problems in the counting hierarchy. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 362-373, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)


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@InProceedings{fournier_et_al:LIPIcs.STACS.2012.362,
  author =	{Fournier, Herv\'{e} and Malod, Guillaume and Mengel, Stefan},
  title =	{{Monomials in arithmetic circuits: Complete problems in the counting hierarchy}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{362--373},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-35-4},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{14},
  editor =	{D\"{u}rr, Christoph and Wilke, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.362},
  URN =		{urn:nbn:de:0030-drops-34240},
  doi =		{10.4230/LIPIcs.STACS.2012.362},
  annote =	{Keywords: arithmetic circuits, counting problems, polynomials}
}
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