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Book ChapterDOI

Arithmetic circuit lower bounds via maxrank

TL;DR: In this article, a polynomial coefficient matrix was introduced and the maximum rank of this matrix under variable substitution was identified as a complexity measure for multivariate polynomials.
Abstract: We introduce the polynomial coefficient matrix and identify 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 [9] for d=ω(1). · As our second main result, we show that there is an explicit polynomial on n variables and degree at most $\frac{n}{2}$ for which any depth-3 circuit C of product dimension at most $\frac{n}{10}$ (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 [14]. Diagonal circuits are of product dimension 1. · We prove a nΩ(logn) 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 [11]. · 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 [7].
Citations
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Book ChapterDOI
25 Aug 2014
TL;DR: In this paper, it was shown that over fixed-size finite fields, there is no depth reduction technique that can be used to compute all the n O(1)-variate and n-degree polynomials in VP by depth 3 circuits of size 2 o(n).
Abstract: In a surprising recent result, Gupta et al. [GKKS13b] have proved that over ℚ any n O(1)-variate and n-degree polynomial in VP can also be computed by a depth three ∑ ∏ ∑ circuit of size \(2^{O(\sqrt{n} \log^{3/2}n)}\) . Over fixed-size finite fields, Grigoriev and Karpinski proved that any ∑ ∏ ∑ circuit that computes the determinant (or the permanent) polynomial of a n×n matrix must be of size 2Ω(n). In this paper, for an explicit polynomial in VP (over fixed-size finite fields), we prove that any ∑ ∏ ∑ circuit computing it must be of size 2Ω(nlogn). The explicit polynomial that we consider is the iterated matrix multiplication polynomial of n generic matrices of size n×n. The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the n O(1)-variate and n-degree polynomials in VP by depth 3 circuits of size 2 o(nlogn). The result of [GK98] can only rule out such a possibility for ∑ ∏ ∑ circuits of size 2 o(n).

5 citations

Posted Content
TL;DR: In this article, Kumar et al. proved an exponential lower bound for the number of variables with $+$ gates as a parent in an proper sub-formula, and showed that the same lower bound holds for the permanent polynomial.
Abstract: We study limitations of polynomials computed by depth two circuits built over read-once polynomials (ROPs) and depth three syntactically multi-linear formulas. We prove an exponential lower bound for the size of the $\Sigma\Pi^{[N^{1/30}]}$ arithmetic circuits built over syntactically multi-linear $\Sigma\Pi\Sigma^{[N^{8/15}]}$ arithmetic circuits computing a product of variable disjoint linear forms on $N$ variables. We extend the result to the case of $\Sigma\Pi^{[N^{1/30}]}$ arithmetic circuits built over ROPs of unbounded depth, where the number of variables with $+$ gates as a parent in an proper sub formula is bounded by $N^{1/2+1/30}$. We show that the same lower bound holds for the permanent polynomial. Finally we obtain an exponential lower bound for the sum of ROPs computing a polynomial in ${\sf VP}$ defined by Raz and Yehudayoff. Our results demonstrate a class of formulas of unbounded depth with exponential size lower bound against the permanent and can be seen as an exponential improvement over the multilinear formula size lower bounds given by Raz for a sub-class of multi-linear and non-multi-linear formulas. Our proof techniques are built on the one developed by Raz and later extended by Kumar et. al.\cite{KMS13} and are based on non-trivial analysis of ROPs under random partitions. Further, our results exhibit strengths and limitations of the lower bound techniques introduced by Raz\cite{Raz04a}.

3 citations

Book ChapterDOI
03 Aug 2017
TL;DR: The notion of fixed parameter tractability is defined and it is shown that there are families of polynomials of degree k that cannot be computed by homogeneous depth four \(\varSigma \varPi ^{\sqrt{k}}\varS Sigma \var Pi ^{k}\) circuits, implying that there is no parameterized depth reduction for circuits of size f(k)n^{O(1) such that the resulting depth four circuit is homogeneous.
Abstract: In this article we initiate the study of polynomials parameterized by degree by arithmetic circuits of small syntactic degree. We define the notion of fixed parameter tractability and show that there are families of polynomials of degree k that cannot be computed by homogeneous depth four \(\varSigma \varPi ^{\sqrt{k}}\varSigma \varPi ^{\sqrt{k}}\) circuits. Our result implies that there is no parameterized depth reduction for circuits of size \(f(k)n^{O(1)}\) such that the resulting depth four circuit is homogeneous.

2 citations

01 Jan 2016
TL;DR: Lower bounds imply deterministic npoly(logn) black-box identity testing algorithms for the above classes of arithmetic circuits.
Abstract: The power symmetric polynomial on n variables of degree d is defined as pd(x1, . . . , xn) = x d 1 + · · · + xn. We study polynomials that are expressible as a sum of powers of homogenous linear projections of power symmetric polynomials. These form a subclass of polynomials computed by depth five circuits with summation and powering gates (i.e., ∑∧∑∧∑ circuits). We show 2Ω(n) size lower bounds for x1 · · ·xn against the following models: • Depth five ∑∧∑≤n∧≥21∑ arithmetic circuits where the bottom ∑ gate is homogeneous; • Depth four ∑∧∑≤n∧ arithmetic circuits. Together with the ideas in [Forbes, FOCS 2015] our lower bounds imply deterministic npoly(logn) black-box identity testing algorithms for the above classes of arithmetic circuits. Our technique uses a measure that involves projecting the partial derivative space of the given polynomial to its multilinear subspace and then setting a subset of variables to 0.

2 citations

Journal ArticleDOI
TL;DR: It is proved that over fixed-size fields there is no depth reduction technique that can be used to compute all the n O ( 1 ) -variate and n-degree polynomials in VP by depth 3 circuits of size 2 o ( n log ⁡ n).
Abstract: In a surprising recent result, Gupta–Kamath–Kayal–Saptharishi have proved that over Q any n O ( 1 ) -variate and n-degree polynomial in VP can also be computed by a depth three ΣΠΣ circuit of size 2 O ( n log 3 / 2 ⁡ n ) . 2 Over fixed-size finite fields, Grigoriev and Karpinski proved that any ΣΠΣ circuit that computes the determinant (or the permanent) polynomial of a n × n matrix must be of size 2 Ω ( n ) . In this paper, for an explicit polynomial in VP (over fixed-size finite fields), we prove that any ΣΠΣ circuit computing it must be of size 2 Ω ( n log ⁡ n ) . The explicit polynomial that we consider is the iterated matrix multiplication polynomial of n generic matrices of size n × n . The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the n O ( 1 ) -variate and n-degree polynomials in VP by depth 3 circuits of size 2 o ( n log ⁡ n ) . The result of Grigoriev and Karpinski can only rule out such a possibility for ΣΠΣ circuits of size 2 o ( n ) . We also give an example of an explicit polynomial ( NW n , ϵ ( X ) ) in VNP (which is not known to be in VP), for which any ΣΠΣ circuit computing it (over fixed-size fields) must be of size 2 Ω ( n log ⁡ n ) . The polynomial we consider is constructed from the combinatorial design of Nisan and Wigderson, and is closely related to the polynomials considered in many recent papers (by Kayal–Saha–Saptharishi, Kayal–Limaye–Saha–Srinivasan, and Kumar–Saraf), where strong depth 4 circuit size lower bounds are shown.

1 citations

References
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Journal ArticleDOI
TL;DR: It is shown that the PSPACE upper bounds cannot be substantially improved without a breakthrough on long standing open problems: the square-root sum problem and an arithmetic circuit decision problem that captures P-time on the unit-cost rational arithmetic RAM model.
Abstract: We define Recursive Markov Chains (RMCs), a class of finitely presented denumerable Markov chains, and we study algorithms for their analysis. Informally, an RMC consists of a collection of finite-state Markov chains with the ability to invoke each other in a potentially recursive manner. RMCs offer a natural abstract model for probabilistic programs with procedures. They generalize, in a precise sense, a number of well-studied stochastic models, including Stochastic Context-Free Grammars (SCFG) and Multi-Type Branching Processes (MT-BP).We focus on algorithms for reachability and termination analysis for RMCs: what is the probability that an RMC started from a given state reaches another target state, or that it terminatesq These probabilities are in general irrational, and they arise as (least) fixed point solutions to certain (monotone) systems of nonlinear equations associated with RMCs. We address both the qualitative problem of determining whether the probabilities are 0, 1 or in-between, and the quantitative problems of comparing the probabilities with a given bound, or approximating them to desired precision.We show that all these problems can be solved in PSPACE using a decision procedure for the Existential Theory of Reals. We provide a more practical algorithm, based on a decomposed version of multi-variate Newton's method, and prove that it always converges monotonically to the desired probabilities. We show this method applies more generally to any monotone polynomial system. We obtain polynomial-time algorithms for various special subclasses of RMCs. Among these: for SCFGs and MT-BPs (equivalently, for 1-exit RMCs) the qualitative problem can be solved in P-time; for linearly recursive RMCs the probabilities are rational and can be computed exactly in P-time.We show that our PSPACE upper bounds cannot be substantially improved without a breakthrough on long standing open problems: the square-root sum problem and an arithmetic circuit decision problem that captures P-time on the unit-cost rational arithmetic RAM model. We show that these problems reduce to the qualitative problem and to the approximation problem (to within any nontrivial error) for termination probabilities of general RMCs, and to the quantitative decision problem for termination (extinction) of SCFGs (MT-BPs).

632 citations

Book
09 Dec 2010
TL;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

Proceedings ArticleDOI
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

Proceedings ArticleDOI
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

Journal ArticleDOI
TL;DR: This paper proves quadratic lower bounds for depth-3 arithmetic circuits over fields of characteristic zero for the elementary symmetric functions, the (trace of) iterated matrix multiplication, and the determinant, and gives new shorter formulae of constant depth for the Elementary symmetrical functions.
Abstract: In this paper we prove quadratic lower bounds for depth-3 arithmetic circuits over fields of characteristic zero. Such bounds are obtained for the elementary symmetric functions, the (trace of) iterated matrix multiplication, and the determinant. As corollaries we get the first nontrivial lower bounds for computing polynomials of constant degree, and a gap between the power of depth-3 arithmetic circuits and depth-4 arithmetic circuits. We also give new shorter formulae of constant depth for the elementary symmetric functions.¶The main technical contribution relates the complexity of computing a polynomial in this model to the wealth of partial derivatives it has on every affine subspace of small co-dimension. Lower bounds for related models utilize an algebraic analog of the Neciporuk lower bound on Boolean formulae.

141 citations