Demystifying the border of depth-3 algebraic circuits

Dutta, Pranjal; Dwivedi, Prateek; Saxena, Nitin

doi:10.1109/focs52979.2021.00018

Cited by 5 publications

(2 citation statements)

References 68 publications

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“…Note that Question 1.17 is linked to the question of understanding commROABP(n, d, w) and ROABP[∀](n, d, w) in Question 1.15. Also, answering this question in the affirmative is similar in spirit to the recent "de-bordering" results due to Dutta et al[DDS21]. They proved that the border of constant top fan-in depth three circuits is contained in the class VBP.…”

supporting

confidence: 79%

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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confidence: 79%

On Finer Separations between Subclasses of Read-once Oblivious ABPs

Ramya¹,

Tengse²

2022

Preprint

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“…Forbes [For16] (see also Bläser, Dörfler, and Ikenmeyer [BDI21]) observed that exact and border complexity are equivalent for read-one oblivious algebraic branching programs. Dutta, Dwivedi, and Saxena [DDS21a] recently showed that polynomials in the border of depth-three circuits of bounded top fan-in can be computed exactly by small algebraic branching programs. However, for classes like VP and VNP (the algebraic analogues of P and NP), it is not clear how they relate to their closure.…”

Section: The Complexity Of Idealsmentioning

confidence: 99%

Ideals, Determinants, and Straightening: Proving and Using Lower Bounds for Polynomial Ideals

Andrews,

Forbes

2021

Preprint

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We show that any nonzero polynomial in the ideal generated by the r × r minors of an n × n matrix X can be used to efficiently approximate the determinant. Specifically, for any nonzero polynomial f in this ideal, we construct a small depth-three f -oracle circuit that approximates the Θ(r 1/3 ) × Θ(r 1/3 ) determinant in the sense of border complexity. For many classes of algebraic circuits, this implies that every nonzero polynomial in the ideal generated by r × r minors is at least as hard to approximately compute as the Θ(r 1/3 ) × Θ(r 1/3 ) determinant. We also prove an analogous result for the Pfaffian of a 2n × 2n skew-symmetric matrix and the ideal generated by Pfaffians of 2r × 2r principal submatrices.This answers a recent question of Grochow [Gro20, Conjecture 6.3] about complexity in polynomial ideals in the setting of border complexity. Leveraging connections between the complexity of polynomial ideals and other questions in algebraic complexity, our results provide a generic recipe that allows lower bounds for the determinant to be applied to other problems in algebraic complexity. We give several such applications, two of which are highlighted below.• We prove new lower bounds for the Ideal Proof System of Grochow and Pitassi. Specifically, we give super-polynomial lower bounds for refutations computed by low-depth circuits. This extends the recent breakthrough low-depth circuit lower bounds of Limaye, Srinivasan, and Tavenas [LST21] to the setting of proof complexity. Moreover, we show that for many natural circuit classes, the approximative proof complexity of our hard instance is governed by the approximative circuit complexity of the determinant.

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Separated borders: Exponential-gap fanin-hierarchy theorem for approximative depth-3 circuits

Dutta

Saxena

2022

2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)

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Demystifying the border of depth-3 algebraic circuits

Cited by 5 publications

References 68 publications

On Finer Separations between Subclasses of Read-once Oblivious ABPs

On Finer Separations between Subclasses of Read-once Oblivious ABPs

Ideals, Determinants, and Straightening: Proving and Using Lower Bounds for Polynomial Ideals

Separated borders: Exponential-gap fanin-hierarchy theorem for approximative depth-3 circuits

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