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Dive into the research topics where Neeraj Kayal is active.

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Featured researches published by Neeraj Kayal.


foundations of computer science | 2013

Arithmetic Circuits: A Chasm at Depth Three

Ankit Gupta; Pritish Kamath; Neeraj Kayal; Ramprasad Saptharishi

We show that, over Q, if an n-variate polynomial of degree d = n<sup>O(1)</sup> is computable by an arithmetic circuit of size s (respectively by an arithmetic branching program of size s) then it can also be computed by a depth three circuit (i.e. a ΣΠΣ-circuit) of size exp(O(√(d log n log d log s))) (respectively of size exp(O(√(d log n log s))). In particular this yields a ΣΠΣ circuit of size exp(O(√(d log d))) computing the d × d determinant Det<sub>d</sub>. It also means that if we can prove a lower bound of exp(omega(√(d log d))) on the size of any ΣΠΣ-circuit computing the d × d permanent Perm<sub>d</sub> then we get super polynomial lower bounds for the size of any arithmetic branching program computing Perm<sub>d</sub>. We then give some further results pertaining to derandomizing polynomial identity testing and circuit lower bounds. The ΣΠΣ circuits that we construct have the property that (some of) the intermediate polynomials have degree much higher than d. Indeed such a counterintuitive construction is unavoidable - it is known that in any ΣΠΣ circuit C computing either Det<sub>d</sub> or Perm_d, if every multiplication gate has fanin at most d (or any constant multiple thereof) then C must have size at least exp(Ω(d)).


foundations of computer science | 2009

Blackbox Polynomial Identity Testing for Depth 3 Circuits

Neeraj Kayal; Shubhangi Saraf

We study depth three arithmetic circuits with bounded top fanin. We give the first deterministic polynomial time blackbox identity test for depth three circuits with bounded top fanin over the field of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main technical result is a structural theorem for depth three circuits with bounded top fanin that compute the zero polynomial. In particular we show that if a circuit C with real coefficients is simple, minimal and computes the zero polynomial, then the rank of C can be upper bounded by a function only of the top fanin. This proves a weak form of a conjecture of Dvir and Shpilka (STOC 2005) on the structure of identically zero depth three arithmetic circuits. Our blackbox identity test follows from this structural theorem by combining it with a construction of Karnin and Shpilka (CCC 2008). Our proof of the structure theorem exploits the geometry of finite point sets in R^n. We identify the linear forms appearing in the circuit C with points in R^n. We then show how to apply high dimensional versions of the Sylvester--Gallai Theorem, a theorem from incidence-geometry, to identify a special linear form appearing in C, such that on the subspace where the linear form vanishes, C restricts to a simpler circuit computing the zero polynomial. This allows us to build an inductive argument bounding the rank of our circuit. While the utility of such theorems from incidence geometry for identity testing has been hinted at before, our proof is the first to develop the connection fully and utilize it effectively.


Journal of the ACM | 2014

Approaching the Chasm at Depth Four

Ankit Gupta; Pritish Kamath; Neeraj Kayal; Ramprasad Saptharishi

Agrawal-Vinay [AV08] and Koiran [Koi12] have recently shown that an exp(ω(√n log2 n)) lower bound for depth four homogeneous circuits computing the permanent with bottom layer of × gates having fanin bounded by √n translates to super-polynomial lower bound for general arithmetic circuits computing the permanent. Motivated by this, we examine the complexity of computing the permanent and determinant via such homogeneous depth four circuits with bounded bottom fanin. We show here that any homogeneous depth four arithmetic circuit with bottom fanin bounded by √n computing the permanent (or the determinant) must be of size exp(Ω(√n)).


symposium on the theory of computing | 2014

A super-polynomial lower bound for regular arithmetic formulas

Neeraj Kayal; Chandan Saha; Ramprasad Saptharishi

We consider arithmetic formulas consisting of alternating layers of addition (+) and multiplication (×) gates such that the fanin of all the gates in any fixed layer is the same. Such a formula Φ which additionally has the property that its formal/syntactic degree is at most twice the (total) degree of its output polynomial, we refer to as a regular formula. As usual, we allow arbitrary constants from the underlying field F on the incoming edges to a + gate so that a + gate can in fact compute an arbitrary F-linear combination of its inputs. We show that there is an (n2 + 1)-variate polynomial of degree 2n in VNP such that any regular formula computing it must be of size at least nΩ(log n). Along the way, we examine depth four (ΣΠΣΠ) regular formulas wherein all multiplication gates in the layer adjacent to the inputs have fanin a and all multiplication gates in the layer adjacent to the output node have fanin b. We refer to such formulas as ΣΠ[b]ΣΠ[a]-formulas. We show that there exists an n2-variate polynomial of degree n in VNP such that any ΣΠ[O(√n)]ΣΠ[√n]-formula computing it must have top fan-in at least 2Ω(√n·log n). In comparison, Tavenas [Tav13] has recently shown that every nO(1)-variate polynomial of degree n in VP admits a ΣΠ[O(√n)]ΣΠ[√n]-formula of top fan-in 2O(√n·log n). This means that any further asymptotic improvement in our lower bound for such formulas (to say 2ω(√n log n)) will imply that VP is different from VNP.


foundations of computer science | 2014

An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

Neeraj Kayal; Nutan Limaye; Chandan Saha; Srikanth Srinivasan

We show here a 2<sup>Ω(√d·log N)</sup> size lower bound for homogeneous depth four arithmetic formulas. That is, we give an explicit family of polynomials of degree d on N variables (with N = d<sup>3</sup> in our case) with 0, 1-coefficients such that for any representation of a polynomial f in this family of the form f = Σ<sub>i</sub> Π<sub>j</sub> Q<sub>ij</sub>, where the Qijs are homogeneous polynomials (recall that a polynomial is said to be homogeneous if all its monomials have the same degree), it must hold that Σi,j (Number of monomials of Q<sub>ij</sub>) ≥ 2<sup>Ω(√d·log N)</sup>. The above mentioned family, which we refer to as the NisanWigderson design-based family of polynomials, is in the complexity class VNP. Our work builds on the recent lower bound results [1], [2], [3], [4], [5] and yields an improved quantitative bound as compared to the quasi-polynomial lower bound of [6] and the N<sup>Ω(log log N)</sup> lower bound in the independent work of [7].


Foundations and Trends in Theoretical Computer Science | 2011

Partial Derivatives in Arithmetic Complexity and Beyond

Xi Chen; Neeraj Kayal; Avi Wigderson

How complex is a given multivariate polynomial? The main point of this survey is that one can learn a great deal about the structure and complexity of polynomials by studying (some of) their partial derivatives. The bulk of the survey shows that partial derivatives provide essential ingredients in proving both upper and lower bounds for computing polynomials by a variety of natural arithmetic models. We will also see applications which go beyond computational complexity, where partial derivatives provide a wealth of structural information about polynomials (including their number of roots, reducibility and internal symmetries), and help us solve various number theoretic, geometric, and combinatorial problems.


symposium on the theory of computing | 2012

Affine projections of polynomials: extended abstract

Neeraj Kayal

An m-variate polynomial f is said to be an affine projection of some n-variate polynomial g if there exists an nm matrix A and an n-dimensional vector b such that f(x)=g(Ax+b). In other words, if f can be obtained by replacing each variable of g by an affine combination of the variables occurring in f, then it is said to be an affine projection of g. Some well known problems (such as the determinant versus permanent and matrix multiplication for example) are instances of this problem. Given f and g can we determine whether f is an affine projection of g? The intention of this paper is to understand the complexity of the corresponding computational problem: given polynomials f and g find A and b such that f=g(Ax+b), if such an (Ab) exists. We first show that this is an NP-hard problem. We then focus our attention on instances where g is a member of some fixed, well known family of polynomials so that the input consists only of the polynomial f(x) having m variables and degree d. We consider the situation where f(x) is given to us as a blackbox (i.e. for any point aFm we can query the blackbox and obtain f(a) in one step) and devise randomized algorithms with running time poly(mnd) in the following special cases. Firstly where g is the Permanent (respectively the Determinant) of an nxn matrix and A is of rank n2. Secondly where g is the sum of powers polynomial (respectively the sum of products polynomial), and A is a random matrix of the appropriate dimensions (also d should not be too small).


conference on computational complexity | 2009

The Complexity of the Annihilating Polynomial

Neeraj Kayal

Let F be a field and f_1, ..., f_k in F[x_1, ..., x_n] be a set of k polynomials of degree d in n variables over the field F. These polynomials are said to be algebraically dependent if there exists a nonzero k-variate polynomial A(t_1, ..., t_k) in F[t_1, ..., t_k] such that A(f_1, ..., f_k) = 0. A is then called an (f_1, ..., f_k)-annihilating polynomial. Within computer science, the notion of algebraic dependence was used in Dvir, Gabizon and Wigderson to construct explicit deterministic extractors from low-degree polynomial sources. They also observed that given (f_1, ..., f_k) as arithmetic circuits, there exists an efficient randomized algorithm for testing their algebraic independence. The problems of determining good bounds on the degree of the annihilating polynomial and of computing it explicitly were posed as open questions. We solve the two posed problems in the following way: ≫≫ We give closely matching upper and lower bounds for the degree of the annihilating polynomial. ≫≫ We show that it is NP-hard to decide if A(0, .. ,0) equals zero. Indeed the annihilating polynomial A(t_1, .., t_k)


conference on computational complexity | 2015

Lower bounds for depth three arithmetic circuits with small bottom fanin

Neeraj Kayal; Chandan Saha

does not even admit a small circuit representation unless the polynomial hierarchy collapses. This then, to the best of our knowledge, is the only natural computational problem where determining the existence of an object (the annihilating polynomial in our case) can be done efficiently but the actual computation of the object is provably hard.


symposium on theoretical aspects of computer science | 2016

Separation Between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth Three Circuits

Neeraj Kayal; Vineet Nair; Chandan Saha

Shpilka and Wigderson [22] had posed the problem of proving exponential lower bounds for (nonhomogeneous) depth three arithmetic circuits with bounded bottom fanin over a field F of characteristic zero. We resolve this problem by proving a [EQUATION] lower bound for (nonhomogeneous) depth three arithmetic circuits with bottom fanin at most τ computing an explicit N-variate polynomial of degree d over F. Meanwhile, Nisan and Wigderson [18] had posed the problem of proving superpolynomial lower bounds for homogeneous depth five arithmetic circuits. Over fields of characteristic zero, we show a lower bound of [EQUATION] for homogeneous depth five circuits (resp. also for depth three circuits) with bottom fanin at most Nμ, for any fixed μ < 1. This resolves the problem posed by Nisan and Wigderson only partially because of the added restriction on the bottom fanin (a general homogeneous depth five circuit has bottom fanin at most N).

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Chandan Saha

Indian Institute of Technology Kanpur

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Ankit Gupta

Chennai Mathematical Institute

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Ramprasad Saptharishi

Chennai Mathematical Institute

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Pritish Kamath

Massachusetts Institute of Technology

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Nitin Saxena

Indian Institute of Technology Kanpur

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Sébastien Tavenas

École normale supérieure de Lyon

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Nutan Limaye

Indian Institute of Technology Bombay

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Srikanth Srinivasan

Indian Institute of Technology Bombay

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