A Quadratic Lower Bound for Algebraic Branching Programs

Chatterjee, Prerona; Kumar, Mrinal; She, Adrian; Volk, Ben Lee

doi:10.4230/lipics.ccc.2020.2

Cited by 3 publications

(2 citation statements)

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“…Although proving strong lower bounds against algebraic circuits seems currently unattainable, even proving lower bounds against ABPs remains a challenging task. In fact, even a superquadratic lower bound against ABPs will be a massive improvement over the state of the art ( [BS83,CKSV20]). A significant amount of work in the area has therefore focused on analysing more structured variants of ABPs which could potentially be easier to tackle.…”

Section: Introductionmentioning

confidence: 99%

On Finer Separations between Subclasses of Read-once Oblivious ABPs

Ramya¹,

Tengse²

2022

Preprint

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Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials as products of univariate polynomials that have matrices as coefficients. In an attempt to understand the landscape of algebraic complexity classes surrounding ROABPs, we study classes of ROABPs based on the algebraic structure of these coefficient matrices. We study connections between polynomials computed by these structured variants of ROABPs and other well-known classes of polynomials (such as depth-three powering circuits, tensor-rank and Waring rank of polynomials).Our main result concerns commutative ROABPs, where all coefficient matrices commute with each other, and diagonal ROABPs, where all the coefficient matrices are just diagonal matrices. In particular, we show a somewhat surprising connection between these models and the model of depth-three powering circuits that is related to the Waring rank of polynomials. We show that if the dimension of partial derivatives captures Waring rank up to polynomial factors, then the model of diagonal ROABPs efficiently simulates the seemingly more expressive model of commutative ROABPs. Further, a commutative ROABP that cannot be efficiently simulated by a diagonal ROABP will give an explicit polynomial that gives a super-polynomial separation between dimension of partial derivatives and Waring rank.Our proof of the above result builds on the results of Marinari, Möller andMora (1993), andStetter (1995), that characterise rings of commuting matrices in terms of polynomials that have small dimension of partial derivatives. The algebraic structure of the coefficient matrices of these ROABPs plays a crucial role in our proofs.

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Section: Introductionmentioning

confidence: 99%

On Finer Separations between Subclasses of Read-once Oblivious ABPs

Ramya¹,

Tengse²

2022

Preprint

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“…Further, skew arithmetic circuits are known to characterize the complexity of determinant [38]. Despite their simplicity compared to arithmetic circuits, the best known lower bound for size of ABPs is only quadratic [21,10]. Even with the restriction of syntactic multilinearity, the best known size lower bound for ABPs is only quadratic [16].…”

Section: Introductionmentioning

confidence: 99%

Limitations of Sums of Bounded-Read Formulas

Ghosal¹,

Rao²

2020

Preprint

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Proving super polynomial size lower bounds for various classes of arithmetic circuits computing explicit polynomials is a very important and challenging task in algebraic complexity theory. We study representation of polynomials as sums of weaker models such as read once formulas (ROFs) and read once oblivious algebraic branching programs (ROABPs). We prove:(1) An exponential separation between sum of ROFs and read-k formulas for some constant k.(2) A sub-exponential separation between sum of ROABPs and syntactic multilinear ABPs.Our results are based on analysis of the partial derivative matrix under different distributions. These results highlight richness of bounded read restrictions in arithmetic formulas and ABPs.Finally, we consider a generalization of multilinear ROABPs known as strict-interval ABPs defined in [Ramya-Rao, MFCS2019]. We show that strict-interval ABPs are equivalent to ROABPs upto a polynomial size blow up. In contrast, we show that interval formulas are different from ROFs and also admit depth reduction which is not known in the case of strict-interval ABPs.

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On Finer Separations Between Subclasses of Read-Once Oblivious ABPs

Ramya,

Tengse

2022

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A Quadratic Lower Bound for Algebraic Branching Programs

Cited by 3 publications

References 0 publications

On Finer Separations between Subclasses of Read-once Oblivious ABPs

On Finer Separations between Subclasses of Read-once Oblivious ABPs

Limitations of Sums of Bounded-Read Formulas

On Finer Separations Between Subclasses of Read-Once Oblivious ABPs

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