Unifying Known Lower Bounds via Geometric Complexity Theory

Grochow, Joshua A.

doi:10.1109/ccc.2014.35

Cited by 5 publications

(8 citation statements)

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“…(32), equals the r.h.s. of (37). This is because u · X c and u ⋄ X c coincide if we think of u as a generic element of G (rather thanḠ), whence det(u) = 1.…”

Section: Consider the Representation Morphismmentioning

confidence: 95%

“…Indeed, all known lower bounds for the exact computation of the permanent also hold for infinitesimally close approximation; eg. see [59,37]. Hence, Theorem 1.7 may also explain why the VNP ⊆ VP ws conjecture in [91] has turned out to be so difficult.…”

Section: Other Extensions In Positive Characteristicmentioning

confidence: 98%

“…LetĀ denote the column vector of length e whose s-the entry, for s ≤ e, is A gs (v, x) = (g s ⋄ X c )(v, x). Then, by (37),…”

Section: Consider the Representation Morphismmentioning

confidence: 99%

“…. , x n , and with multiple outputs that compute the polynomials β µ (v, x)'s in (37). The top (output) gates of C ′ are all addition gates.…”

Section: Consider the Representation Morphismmentioning

confidence: 99%

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Geometric complexity theory V: Efficient algorithms for Noether normalization

Mulmuley

2016

J. Amer. Math. Soc.

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We study a basic algorithmic problem in algebraic geometry, which we call NNL, of constructing a normalizing map as per Noether's Normalization Lemma. For general explicit varieties, as formally defined in this paper, we give a randomized polynomial-time Monte Carlo algorithm for this problem. For some interesting cases of explicit varieties, we give deterministic quasi-polynomial time algorithms. These may be contrasted with the standard EXPSPACE-algorithms for these problems in computational algebraic geometry.In particular, we show that:(1) The categorical quotient for any finite dimensional representation V of SL m , with constant m, is explicit in characteristic zero.(2) NNL for this categorical quotient can be solved deterministically in time quasipolynomial in the dimension of V .(3) The categorical quotient of the space of r-tuples of m×m matrices by the simultaneous conjugation action of SL m is explicit in any characteristic.(4) NNL for this categorical quotient can be solved deterministically in time quasipolynomial in m and r in any characteristic p ∈ [2, ⌊m/2⌋].(5) NNL for every explicit variety in zero or large enough characteristic can be solved deterministically in quasi-polynomial time, assuming the hardness hypothesis for the permanent in geometric complexity theory.The last result leads to a geometric complexity theory approach to put NNL for every explicit variety in P.

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“…(32), equals the r.h.s. of (37). This is because u · X c and u ⋄ X c coincide if we think of u as a generic element of G (rather thanḠ), whence det(u) = 1.…”

Section: Consider the Representation Morphismmentioning

confidence: 95%

Section: Other Extensions In Positive Characteristicmentioning

confidence: 98%

“…LetĀ denote the column vector of length e whose s-the entry, for s ≤ e, is A gs (v, x) = (g s ⋄ X c )(v, x). Then, by (37),…”

Section: Consider the Representation Morphismmentioning

confidence: 99%

“…. , x n , and with multiple outputs that compute the polynomials β µ (v, x)'s in (37). The top (output) gates of C ′ are all addition gates.…”

Section: Consider the Representation Morphismmentioning

confidence: 99%

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Geometric complexity theory V: Efficient algorithms for Noether normalization

Mulmuley

2016

J. Amer. Math. Soc.

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“…The author would also like to thank Allan Borodin for his support, both material and otherwise, part of which came by way of Borodin's NSERC Grant #482671. An extended abstract of this paper has appeared as Grochow (2014). Proof.…”

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confidence: 99%

Unifying Known Lower Bounds via Geometric Complexity Theory

Grochow

2015

comput. complex.

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Abstract. We show that most algebraic circuit lower bounds and relations between lower bounds naturally fit into the representationtheoretic framework suggested by geometric complexity theory (GCT), including: the partial derivatives technique (Nisan-Wigderson), the results of Razborov and Smolensky on AC 0 [p], multilinear formula and circuit size lower bounds (Raz et al.), the degree bound (Strassen, BaurStrassen), the connected components technique (Ben-Or, Steele-Yao), depth 3 algebraic circuit lower bounds over finite fields (GrigorievKarpinski), lower bounds on permanent versus determinant (MignonRessayre, Landsberg-Manivel-Ressayre), lower bounds on matrix multiplication (Bürgisser-Ikenmeyer, Landsberg-Ottaviani) (these last two were already known to fit into GCT), the chasms at depth 3 and 4 (Gupta-Kayal-Kamath-Saptharishi, Agrawal-Vinay, Koiran, Tavenas), matrix rigidity (Valiant) and others. That is, the original proofs, with what is often just a little extra work, already provide representation-theoretic obstructions in the sense of GCT for their respective lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a natural unification of known results and broad generalization of known techniques. It also shows that the framework of GCT is at least as powerful as previous methods, and gives many new proofs-of-concept that GCT can indeed provide significant asymptotic lower bounds. This new viewpoint also opens up the possibility of fruitful two-way interactions between previous results and the new methods of GCT; we provide several concrete suggestions of such interactions. For example, the representation-theoretic viewpoint of GCT 394 Grochow cc 24 (2015) naturally provides new properties to consider in the search for new lower bounds.Keywords. Lower bounds, circuit complexity, algebraic complexity, geometric complexity theory, representation theory, algebraic geometry. Subject classification. 68Q17, 14L30, 13A50.Parts of this paper are written in a survey style in order to make it accessible to a wider audience. In particular, items marked Example or Fact are included only for expository purposes, and we make no claim to originality for those.This paper presupposes no knowledge of representation theory on the part of the reader. In fact, we use previous lower bounds together with our new viewpoint to motivate the use and definitions of representation theory and algebraic geometry in complexity theory.

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The method of shifted partial derivatives cannot separate the permanent from the determinant

et al. 2017

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Abstract. The method of shifted partial derivatives introduced in [9,7] was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this method alone cannot prove that the padded permanent ℓ n−m perm m cannot be realized inside the GL n 2 -orbit closure of the determinant detn when n > 2m 2 + 2m. Our proof relies on several simple degenerations of the determinant polynomial, Macaulay's theorem that gives a lower bound on the growth of an ideal, and a lower bound estimate from [7] regarding the shifted partial derivatives of the determinant.

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Unifying Known Lower Bounds via Geometric Complexity Theory

Cited by 5 publications

References 71 publications

Geometric complexity theory V: Efficient algorithms for Noether normalization

Geometric complexity theory V: Efficient algorithms for Noether normalization

Unifying Known Lower Bounds via Geometric Complexity Theory

The method of shifted partial derivatives cannot separate the permanent from the determinant

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