Geometric complexity theory V: Efficient algorithms for Noether normalization

Mulmuley, Ketan

doi:10.1090/jams/864

Cited by 36 publications

(31 citation statements)

References 90 publications

(298 reference statements)

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“…Yet, the hope of proving lower bounds via optimization methods is a fascinating (and possibly achievable) one. This agenda of hoping to shed light on the PIT problem by the study of invariant theoretic questions was formulated in the fifth paper of the Geometric Complexity Theory (GCT) series [Mul12,Mul17].…”

Section: Some Unexpected Applications and Connectionsmentioning

confidence: 99%

Towards a Theory of Non-Commutative Optimization: Geodesic 1st and 2nd Order Methods for Moment Maps and Polytopes

Bürgisser

Franks

Garg

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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This paper initiates a systematic development of a theory of non-commutative optimization, a setting which greatly extends ordinary (Euclidean) convex optimization. It aims to unify and generalize a growing body of work from the past few years which developed and analyzed algorithms for natural geodesically convex optimization problems on Riemannian manifolds that arise from the symmetries of non-commutative groups. More specifically, these are algorithms to minimize the moment map (a non-commutative notion of the usual gradient), and to test membership in moment polytopes (a vast class of polytopes, typically of exponential vertex and facet complexity, which quite magically arise from this a-priori non-convex, non-linear setting).The importance of understanding this very general setting of geodesic optimization, as these works unveiled and powerfully demonstrate, is that it captures a diverse set of problems, many non-convex, in different areas of CS, math, and physics. Several of them were solved efficiently for the first time using non-commutative methods; the corresponding algorithms also lead to solutions of purely structural problems and to many new connections between disparate fields.In the spirit of standard convex optimization, we develop two general methods in the geodesic setting, a first order and a second order method, which respectively receive first and second order information on the "derivatives" of the function to be optimized. These in particular subsume all past results. The main technical work, again unifying and extending much of the previous work, goes into identifying the key parameters of the underlying group actions which control convergence to the optimum in each of these methods. These non-commutative analogues of "smoothness" in the commutative case are far more complex, and require significant algebraic and analytic machinery (much existing and some newly developed here). Despite this complexity, the way in which these parameters control convergence in both methods is quite simple and elegant. We also bound these parameters in several general cases.Our work points to intriguing open problems and suggests further research directions. We believe that extending this theory, namely understanding geodesic optimization better, is both mathematically and computationally fascinating; it provides a great meeting place for ideas and techniques from several very different research areas, and promises better algorithms for existing and yet unforeseen applications.

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Section: Some Unexpected Applications and Connectionsmentioning

confidence: 99%

Towards a Theory of Non-Commutative Optimization: Geodesic 1st and 2nd Order Methods for Moment Maps and Polytopes

Bürgisser

Franks

Garg

et al. 2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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“…X that depend on "constantly few" variables (see the reduction of the derandomized Noether normalization problem NNL to blackbox PIT in ref. 4…”

Section: Our Notationmentioning

confidence: 99%

“…The major open question is to bring this complexity down to P; this is christened the "GCT Chasm" in ref. 4, section 11 and has since become a fundamental difficulty common to geometry and complexity theories. It means that we have to discover algebraic properties that are specific to only those polynomials that have small circuit representation.…”

Section: Section 43)mentioning

confidence: 99%

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Bootstrapping variables in algebraic circuits

Agrawal

Ghosh

Saxena

2019

Proc. Natl. Acad. Sci. U.S.A.

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We show that for the blackbox polynomial identity testing (PIT) problem it suffices to study circuits that depend only on the first extremely few variables. One needs only to consider size-s degree-s circuits that depend on the firstlog○c svariables (where c is a constant and composes a logarithm with itself c times). Thus, the hitting-set generator (hsg) manifests a bootstrapping behavior—a partial hsg against very few variables can be efficiently grown to a complete hsg. A Boolean analog, or a pseudorandom generator property of this type, is unheard of. Our idea is to use the partial hsg and its annihilator polynomial to efficiently bootstrap the hsg exponentially w.r.t. variables. This is repeated c times in an efficient way. Pushing the envelope further we show that (i) a quadratic-time blackbox PIT for 6,913-variate degree-s size-s polynomials will lead to a “near”-complete derandomization of PIT and (ii) a blackbox PIT for n-variate degree-s size-s circuits insnδtime, forδ<1/2, will lead to a near-complete derandomization of PIT (in contrast,sntime is trivial). Our second idea is to study depth-4 circuits that depend on constantly many variables. We show that a polynomial-time computable,O(s1.49)-degree hsg for trivariate depth-4 circuits bootstraps to a quasipolynomial time hsg for general polydegree circuits and implies a lower bound that is a bit stronger than that of Kabanets and Impagliazzo [Kabanets V, Impagliazzo R (2003)Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing STOC ’03].

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“…The above problem is interesting as natural questions like explicit construction of the normalization map (in Noether's Normalization Lemma NNL) reduce to the construction of a hitting set of VP (Mulmuley [45]), which was previously known to be only in EXPSPACE [45,44]. This was recently put in PSPACE, over the field C, by Forbes and Shpilka [21].…”

Section: Introductionmentioning

confidence: 99%

Untitled

Guo¹,

Saxena²,

Sinhababu³

2019

Theory of Comput.

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Testing whether a set f of polynomials has an algebraic dependence is a basic problem with several applications. The polynomials are given as algebraic circuits. The complexity of algebraic independence testing is wide open over finite fields (Dvir, Gabizon, Wigderson, FOCS'07). Previously, the best complexity bound known was NP #P (Mittmann, Saxena, Scheiblechner, Trans. AMS 2014). In this article we put the problem in AM ∩ coAM. In particular, dependence testing is unlikely to be NP-hard. Our proof uses methods of algebraic geometry. We estimate the size of the image and the sizes of the preimages of the polynomial map f over the finite field. A gap between the corresponding sizes for independent and for dependent sets of polynomials is utilized in the AM protocols.

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Geometric complexity theory V: Efficient algorithms for Noether normalization

Cited by 36 publications

References 90 publications

Towards a Theory of Non-Commutative Optimization: Geodesic 1st and 2nd Order Methods for Moment Maps and Polytopes

Towards a Theory of Non-Commutative Optimization: Geodesic 1st and 2nd Order Methods for Moment Maps and Polytopes

Bootstrapping variables in algebraic circuits

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