Lower Bounds for Special Cases of Syntactic Multilinear ABPs

Ramya, C.; Rao, B. V. Raghavendra

doi:10.1007/978-3-319-94776-1_58

Cited by 4 publications

(7 citation statements)

References 12 publications

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“…Though the overall idea is simple, the construction requires a lot of bookkeeping of variable orders. Combining Theorem 5 with the lower bound for sum of k-pass smABPs given in [14], we get an exponential lower bound for sum of L-ordered smABPs when k = o(log n). By exploiting a simple structural property of L ordered ABPs, we prove an exponential lower bound for sum of L ordered smABPs even when L is subexponentially small:…”

Section: Models and Resultsmentioning

confidence: 79%

Lower bounds for multilinear bounded order ABPs

Ramya,

Rao

2019

Preprint

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Proving super-polynomial size lower bounds for syntactic multilinear Algebraic Branching Programs(smABPs) computing an explicit polynomial is a challenging problem in Algebraic Complexity Theory. The order in which variables in {x 1 , . . . , x n } appear along a source to sink path in any smABP can be viewed as a permutation in S n . In this article, we consider the following special classes of smABPs where the order of occurrence of variables along a source to sink path is restricted:

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Section: Models and Resultsmentioning

confidence: 79%

Lower bounds for multilinear bounded order ABPs

Ramya,

Rao

2019

Preprint

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“…Kayal et al [19] obtain an exponential separation between the size of ROABPs and depth three multilinear formulas. In [27], an exponential lower bound for the sum of ROABPs computing a polynomial in VP is given. We improve this bound to obtain a super polynomial separation between sum of ROABPs and smABPs: Theorem 2.…”

Section: Models and Results: (1) Sum Of Rofsmentioning

confidence: 99%

“…This polynomial can be expressed as a sum of three ROFs. Later, Ramya and Rao [27] obtain a sub-exponential lower bound against the model of Σ • ROABP computing the polynomial defined by Raz and Yehudayoff [33]. Dvir et al [13] obtain a super-polynomial lower bound on the size of syntactic multilinear formulas computing a polynomial that can be efficiently computed by smABPs.…”

Section: Limitations Of Sums Of Bounded Read Formulasmentioning

confidence: 99%

“…Given that there is no promising approach yet to prove super quadratic size lower bounds for smABPs, it is imperative to consider further structural restrictions on smABPs and formulas to develop finer insights into the difficulty of the problem. Following the works in [27,29,28], we study syntactic multilinear formulas and smABPs with restrictions on the number of reads of variables and the order in which variables appear in a smABP.…”

Section: Introductionmentioning

confidence: 99%

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Limitations of Sums of Bounded-Read Formulas

Ghosal¹,

Rao²

2020

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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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“…Many interesting classes of circuits can be defined by restricting the set of allowed parse trees, both in the commutative and the non-commutative setting. The simplest such class is that of Algebraic Branching Programs (ABP) [20,7,22], whose only parse trees are left-combs, that is, the variables are multiplied sequentially. Lagarde, Malod and Perifel introduced in [17] the class of Unique Parse Tree circuits (UPT), which are circuits computing non-commutative homogeneous (but associative) polynomials such that all monomials are evaluated in the same non-associative way.…”

Section: Parse Treesmentioning

confidence: 99%

Lower Bounds for Arithmetic Circuits via the Hankel Matrix

et al. 2021

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We study the complexity of representing polynomials by arithmetic circuits in both the commutative and the non-commutative settings. To analyse circuits we count their number of parse trees, which describe the non-associative computations realised by the circuit.In the non-commutative setting a circuit computing a polynomial of degree d has at most 2 O(d) parse trees. Previous superpolynomial lower bounds were known for circuits with up to 2 d 1/3− parse trees, for any ε > 0. Our main result is to reduce the gap by showing a superpolynomial lower bound for circuits with just a small defect in the exponent for the total number of parse trees, that is 2 d 1−ε , for any ε > 0. In the commutative setting a circuit computing a polynomial of degree d has at most 2 O(d log d) parse trees. We show a superpolynomial lower bound for circuits with up to 2 d 1/3−ε parse trees, for any ε > 0. When d is polylogarithmic in n, we push this further to up to 2 d 1−ε parse trees. While these two main results hold in the associative setting, our approach goes through a precise understanding of the more restricted setting where multiplication is not associative, meaning that we distinguish the polynomials (xy)z and x(yz). Our first and main conceptual result is a characterization result: we show that the size of the smallest circuit computing a given nonassociative polynomial is exactly the rank of a matrix constructed from the polynomial and called the Hankel matrix. This result applies to the class of all circuits in both commutative and noncommutative settings, and can be seen as an extension of the seminal result of Nisan giving a similar characterization for non-commutative algebraic branching programs. Our key technical contribution is to provide generic lower bound theorems based on analyzing and decomposing the Hankel matrix, from which we derive the results mentioned above.The study of the Hankel matrix also provides a unifying approach for proving lower bounds for polynomials in the (classical) associative setting. We demonstrate this by giving alternative proofs of recent lower bounds as corollaries of our generic lower bound results.

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Lower Bounds for Special Cases of Syntactic Multilinear ABPs

Cited by 4 publications

References 12 publications

Lower bounds for multilinear bounded order ABPs

Lower bounds for multilinear bounded order ABPs

Limitations of Sums of Bounded-Read Formulas

Lower Bounds for Arithmetic Circuits via the Hankel Matrix

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