Waring Rank, Parameterized and Exact Algorithms

Pratt, Kevin

doi:10.1109/focs.2019.00053

Cited by 13 publications

(13 citation statements)

References 50 publications

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“…This is a known fact(see e.g. [Pra19]). We start with a fact about the "symmetry" between D h (g)( 0) and D g (h)( 0) that we will need.…”

Section: Evaluating Derivatives Of Polynomialsmentioning

confidence: 83%

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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“…This is a known fact(see e.g. [Pra19]). We start with a fact about the "symmetry" between D h (g)( 0) and D g (h)( 0) that we will need.…”

Section: Evaluating Derivatives Of Polynomialsmentioning

confidence: 83%

On Finer Separations between Subclasses of Read-once Oblivious ABPs

Ramya¹,

Tengse²

2022

Preprint

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“…In condition (4), the quantity r is known as the "Waring rank" of this polynomial. Pratt [18] presents a history and many interesting results about this classical invariant.…”

Section: Suppose (3)mentioning

confidence: 99%

Recurrence Ranks and Moment Sequences

Cooper¹,

Fickes²

2021

Preprint

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We introduce the "moment rank" and "unitary rank" of numerical sequences, close relatives of linear-recursive order. We show that both parameters can be characterized by a broad set of criteria involving moments of measures, types of recurrence relations, Hankel matrix factorizations, Waring rank, analytic properties of generating functions, and algebraic properties of polynomial ideals. In the process, we solve the "complex finite-atomic" and "integral finite-atomic" moment problems: which sequences arise as the moments of a finite-atomic complex-/integer-valued measures on C?

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“…A series of work has improved the computable function g, see e.g. [13,40,58]. For many special pattern graphs faster algorithms have been found; the most famous example is the k-Clique problem, which can be solved in time O(n kω/3 g(k)) [57] 1 .…”

Section: Introductionmentioning

confidence: 99%

Current Algorithms for Detecting Subgraphs of Bounded Treewidth are Probably Optimal

Bringmann,

Slusallek

2021

Preprint

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Waring Rank, Parameterized and Exact Algorithms

Cited by 13 publications

References 50 publications

On Finer Separations between Subclasses of Read-once Oblivious ABPs

On Finer Separations between Subclasses of Read-once Oblivious ABPs

Recurrence Ranks and Moment Sequences

Current Algorithms for Detecting Subgraphs of Bounded Treewidth are Probably Optimal

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