Arithmetic Circuit Lower Bounds via Maximum-Rank of Partial Derivative Matrices
TL;DR: The polynomial coefficient matrix is introduced and the maximum rank of this matrix under variable substitution is identified as a complexity measure for multivariate polynomials and the techniques used are used to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits.
Abstract: We introduce the polynomial coefficient matrix and identify the maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results:—As our first main result, we prove that any homogeneous depth-3 circuit for computing the product of d matrices of dimension n × n requires Ω(nd − 1/2d) size. This improves the lower bounds in Nisan and Wigderson [1995] for d = ω(1).—As our second main result, we show that there is an explicit polynomial on n variables and degree at most n/2 for which any depth-3 circuit of product dimension at most n/10 (dimension of the space of affine forms feeding into each product gate) requires size 2Ω(n). This generalizes the lower bounds against diagonal circuits proved in Saxena [2008]. Diagonal circuits are of product dimension 1.—We prove a nΩ(log n) lower bound on the size of product-sparse formulas. By definition, any multilinear formula is a product-sparse formula. Thus, this result extends the known super-polynomial lower bounds on the size of multilinear formulas [Raz 2006].—We prove a 2Ω(n) lower bound on the size of partitioned arithmetic branching programs. This result extends the known exponential lower bound on the size of ordered arithmetic branching programs [Jansen 2008].
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TL;DR: In this article, the Razborov-Rudich natural proofs barrier in Boolean circuit complexity was introduced, in which a certain kind of algebraic proof cannot be used to prove lower bounds against VP if and only if what are called succinct hitting sets exist for VP.
Abstract: We observe that a certain kind of algebraic proof - which covers essentially all known algebraic circuit lower bounds to date - cannot be used to prove lower bounds against VP if and only if what we call succinct hitting sets exist for VP. This is analogous to the Razborov-Rudich natural proofs barrier in Boolean circuit complexity, in that we rule out a large class of lower bound techniques under a derandomization assumption. We also discuss connections between this algebraic natural proofs barrier, geometric complexity theory, and (algebraic) proof complexity.
18 citations
Journal Article•
TL;DR: This work gives an explicit family of polynomials of degree d on N variables with 0, 1-coefficients such that for any representation of a polynomial f in this family of the form f = Σi,j, it must hold that 2Ω(√d·log N) ≥ 2.
Abstract: We show here a $2^{\Omega(\sqrt{d} \cdot \log N)}$ size lower bound for homogeneous depth four arithmetic formulas over fields of characteristic zero That is, we give an explicit family of polynomials of degree $d$ on $N$ variables (with $N = d^3$ in our case) with 0, 1-coefficients such that for any representation of a polynomial $f$ in this family of the form $ f = \sum_{i} \prod_{j} Q_{ij}, $ where the $Q_{ij}$'s are homogeneous polynomials (recall that a polynomial is said to be homogeneous if all its monomials have the same degree), it must hold that $ \sum_{i, j} (\text{number of monomials of~} Q_{ij}) \geq 2^{\Omega (\sqrt{d} \cdot \log N)} $ The abovementioned family, which we refer to as the Nisan--Wigderson design-based family of polynomials, is in the complexity class $\mathsf{VNP}$ Our work builds on recent lower bound results and yields an improved quantitative bound as compared to the quasi-polynomial lower bound of [N Kayal et al, in Symposium on Theory of Computing, ACM, New York, 201
10 citations
01 Jan 2016
TL;DR: The proof techniques are built on the measure developed by Kumar et al.[ICALP 2013] and are based on a non-trivial analysis of ROFs under random partitions and provide more insight into the lower bound techniques introduced by Raz [STOC 2004].
Abstract: We study limitations of polynomials computed by depth two circuits built over read-once formulas (ROFs). In particular,
1. We prove an exponential lower bound for the sum of ROFs computing the 2n-variate polynomial in VP defined by Raz and Yehudayoff [CC,2009].
2. We obtain an exponential lower bound on the size of arithmetic circuits computing sum of products of restricted ROFs of unbounded depth computing the permanent of an n by n matrix. The restriction is on the number of variables with + gates as a parent in a proper sub formula of the ROF to be bounded by sqrt(n). Additionally, we restrict the product fan in to be bounded by a sub linear function. This proves an exponential lower bound for a subclass of possibly non-multilinear formulas of unbounded depth computing the permanent polynomial.
3. We also show an exponential lower bound for the above model against a polynomial in VP.
4. Finally we observe that the techniques developed yield an exponential lower bound on the size of sums of products of syntactically multilinear arithmetic circuits computing a product of variable disjoint linear forms where the bottom sum gate and product gates at the second level have fan in bounded by a sub linear function.
Our proof techniques are built on the measure developed by Kumar et al.[ICALP 2013] and are based on a non-trivial analysis of ROFs under random partitions. Further, our results exhibit strengths and provide more insight into the lower bound techniques introduced by Raz [STOC 2004].
6 citations
TL;DR: A 2(n) lower bound is proved for the sum of ROFs computing the 2n-variate polyno... of polynomials computed by depth-2 circuits built over read-once formulas (ROFs).
Abstract: We study limitations of polynomials computed by depth-2 circuits built over read-once formulas (ROFs). In particular:• We prove a 2Ω(n) lower bound for the sum of ROFs computing the 2n-variate polynomial in VP defined by Raz and Yehudayoff l21r.• We obtain a 2Ω(√n) lower bound on the size of Σ Πln1/15r arithmetic circuits built over restricted ROFs of unbounded depth computing the permanent of an n × n matrix (superscripts on gates denote bound on the fan-in). The restriction is that the number of variables with + gates as a parent in a proper sub formula of the ROF has to be bounded by √n. This proves an exponential lower bound for a subclass of possibly non-multilinear formulas of unbounded depth computing the permanent polynomial.• We also show an exponential lower bound for the above model against a polynomial in VP.• Finally, we observe that the techniques developed yield an exponential lower bound on the size of ΣΠlN1/30r arithmetic circuits built over syntactically multi-linear ΣΠΣlN1/4r arithmetic circuits computing a product of variable disjoint linear forms on N variables, where the superscripts on gates denote bound on the fan-in.Our proof techniques are built on the measure developed by Kumar et al. l14r and are based on a non-trivial analysis of ROFs under random partitions. Further, our results exhibit strengths and provide more insight into the lower bound techniques introduced by Raz l19r.
4 citations
References
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Book•
09 Dec 2010TL;DR: The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what it finds to be the most interesting and accessible research directions, with an emphasis on works from the last two decades.
Abstract: A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions.
The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we discuss the recent lower bounds for multilinear circuits and formulas, the advances in the question of deterministically checking polynomial identities, and the results regarding reconstruction of arithmetic circuits. We do, however, also cover part of the classical works on arithmetic circuits. In order to keep this monograph at a reasonable length, we do not give full proofs of most theorems, but rather try to convey the main ideas behind each proof and demonstrate it, where possible, by proving some special cases.
509 citations
25 Oct 2008
TL;DR: It is shown that proving exponential lower bounds on depth four arithmetic circuits imply exponentialLower bounds for unrestricted depth arithmetic circuits, and that for exponential sized circuits additional depth beyond four does not help.
Abstract: We show that proving exponential lower bounds on depth four arithmetic circuits imply exponential lower bounds for unrestricted depth arithmetic circuits. In other words, for exponential sized circuits additional depth beyond four does not help. We then show that a complete black-box derandomization of identity testing problem for depth four circuits with multiplication gates of small fanin implies a nearly complete derandomization of general identity testing.
227 citations
TL;DR: A new technique is described for obtaining lower bounds on restricted classes of non-monotone arithmetic circuits, based on the linear span of their partial derivatives, for multivariate polynomials.
Abstract: In this paper we describe a new technique for obtaining lower bounds on restricted classes of non-monotone arithmetic circuits. The heart of this technique is a complexity measure for multivariate polynomials, based on the linear span of their partial derivatives. We use the technique to obtain new lower bounds for computing symmetric polynomials (that hold over fields of characteristic zero) and iterated matrix products (that hold for all fields).
196 citations
Journal Article•
TL;DR: In this article, it was shown that a complete black-box derandomization of identity testing problem for depth four arithmetic circuits with multiplication gates of small fanin implies a nearly complete derandomisation of general identity testing.
Abstract: We show that proving exponential lower bounds on depth four arithmetic circuits imply exponential lower bounds for unrestricted depth arithmetic circuits. In other words, for exponential sized circuits additional depth beyond four does not help. We then show that a complete black-box derandomization of identity testing problem for depth four circuits with multiplication gates of small fanin implies a nearly complete derandomization of general identity testing.
181 citations
23 May 1998
TL;DR: It is proved the first exponential lower bound on the size of any depth 3 arithmetic circuit with unbounded fanin computing an explicit function (the determinant) over an arbitrary finite field.
Abstract: We prove the rst exponential lower bound on the size of any depth 3 arithmetic circuit with unbounded fanin computing an explicit function (the determinant) over an arbitrary nite eld. This answers an open problem of N91] and NW95] for the case of nite elds. We intepret here arithmetic circuits in the algebra of polynomials over the given eld. The proof method involves a new argument on the rank of linear functions, and a group symmetry on polynomials vanishing at certain nonsingular matrices, and could be of independent interest.
141 citations