All reports by Author Mrinal Kumar:

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TR20-129
| 5th September 2020
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Mrinal Kumar, Ben Lee Volk#### A Lower Bound on Determinantal Complexity

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TR20-041
| 29th March 2020
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Mrinal Kumar, Ben Lee Volk#### A Polynomial Degree Bound on Defining Equations of Non-rigid Matrices and Small Linear Circuits

Revisions: 2

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TR19-172
| 28th November 2019
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Prasad Chaugule, Mrinal Kumar, Nutan Limaye, Chandra Kanta Mohapatra, Adrian She, Srikanth Srinivasan#### Schur Polynomials do not have small formulas if the Determinant doesn't!

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TR19-170
| 27th November 2019
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Prerona Chatterjee, Mrinal Kumar, Adrian She, Ben Lee Volk#### A Quadratic Lower Bound for Algebraic Branching Programs

Revisions: 3

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TR19-065
| 1st May 2019
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Mrinal Kumar, Ramprasad Saptharishi, Noam Solomon#### Derandomization from Algebraic Hardness: Treading the Borders

Revisions: 3

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TR19-047
| 2nd April 2019
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Mrinal Kumar, Ben Lee Volk#### Lower Bounds for Matrix Factorization

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TR19-037
| 5th March 2019
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Chi-Ning Chou, Mrinal Kumar, Noam Solomon#### Closure of VP under taking factors: a short and simple proof

Revisions: 1

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TR19-019
| 19th February 2019
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Mrinal Kumar, Rafael Mendes de Oliveira, Ramprasad Saptharishi#### Towards Optimal Depth Reductions for Syntactically Multilinear Circuits

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TR18-132
| 17th July 2018
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Mrinal Kumar, Ramprasad Saptharishi, Anamay Tengse#### Near-optimal Bootstrapping of Hitting Sets for Algebraic Circuits

Revisions: 2

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TR18-081
| 20th April 2018
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Abhishek Bhrushundi, Prahladh Harsha, Pooya Hatami, Swastik Kopparty, Mrinal Kumar#### On Multilinear Forms: Bias, Correlation, and Tensor Rank

Revisions: 1

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TR18-068
| 8th April 2018
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Mrinal Kumar#### On top fan-in vs formal degree for depth-3 arithmetic circuits

Revisions: 1

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TR18-052
| 16th March 2018
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Chi-Ning Chou, Mrinal Kumar, Noam Solomon#### Some Closure Results for Polynomial Factorization and Applications

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TR17-124
| 6th August 2017
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Mrinal Kumar, Ben Lee Volk#### An Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits

Revisions: 2

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TR17-028
| 17th February 2017
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Mrinal Kumar#### A quadratic lower bound for homogeneous algebraic branching programs

Revisions: 1

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TR17-009
| 19th January 2017
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Joshua Grochow, Mrinal Kumar, Michael Saks, Shubhangi Saraf#### Towards an algebraic natural proofs barrier via polynomial identity testing

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TR16-137
| 3rd September 2016
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Mrinal Kumar, Ramprasad Saptharishi#### Finer separations between shallow arithmetic circuits

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TR16-096
| 14th June 2016
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Suryajith Chillara, Mrinal Kumar, Ramprasad Saptharishi, V Vinay#### The Chasm at Depth Four, and Tensor Rank : Old results, new insights

Revisions: 2

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TR16-045
| 22nd March 2016
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Michael Forbes, Mrinal Kumar, Ramprasad Saptharishi#### Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity

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TR15-194
| 30th November 2015
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Mrinal Kumar, Shubhangi Saraf#### Arithmetic circuits with locally low algebraic rank

Revisions: 1

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TR15-109
| 1st July 2015
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Mrinal Kumar, Ramprasad Saptharishi#### An exponential lower bound for homogeneous depth-5 circuits over finite fields

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TR15-071
| 23rd April 2015
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Mrinal Kumar, Shubhangi Saraf#### Sums of products of polynomials in few variables : lower bounds and polynomial identity testing

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TR15-047
| 2nd April 2015
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Swastik Kopparty, Mrinal Kumar, Michael Saks#### Efficient indexing of necklaces and irreducible polynomials over finite fields

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TR14-045
| 7th April 2014
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Mrinal Kumar, Shubhangi Saraf#### On the power of homogeneous depth 4 arithmetic circuits

Revisions: 2

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TR13-181
| 20th December 2013
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Mrinal Kumar, Shubhangi Saraf#### Superpolynomial lower bounds for general homogeneous depth 4 arithmetic circuits

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TR13-153
| 8th November 2013
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Mrinal Kumar, Shubhangi Saraf#### The Limits of Depth Reduction for Arithmetic Formulas: It's all about the top fan-in

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TR13-068
| 3rd May 2013
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Mrinal Kumar, Shubhangi Saraf#### Lower Bounds for Depth 4 Homogenous Circuits with Bounded Top Fanin

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TR13-028
| 14th February 2013
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Mrinal Kumar, Gaurav Maheshwari, Jayalal Sarma#### Arithmetic Circuit Lower Bounds via MaxRank

Mrinal Kumar, Ben Lee Volk

The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P = Det(M)$. We prove that the determinantal complexity of the polynomial $\sum_{i = 1}^n x_i^n$ ... more >>>

Mrinal Kumar, Ben Lee Volk

We show that there is a defining equation of degree at most poly(n) for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field $\mathbb{F}$, there is a non-zero $n^2$-variate polynomial $P \in \mathbb{F}(x_{1, 1}, \ldots, x_{n, n})$ of degree ... more >>>

Prasad Chaugule, Mrinal Kumar, Nutan Limaye, Chandra Kanta Mohapatra, Adrian She, Srikanth Srinivasan

Schur Polynomials are families of symmetric polynomials that have been

classically studied in Combinatorics and Algebra alike. They play a central

role in the study of Symmetric functions, in Representation theory [Sta99], in

Schubert calculus [LM10] as well as in Enumerative combinatorics [Gas96, Sta84,

Sta99]. In recent years, they have ...
more >>>

Prerona Chatterjee, Mrinal Kumar, Adrian She, Ben Lee Volk

We show that any Algebraic Branching Program (ABP) computing the polynomial $\sum_{i = 1}^n x_i^n$ has at least $\Omega(n^2)$ vertices. This improves upon the lower bound of $\Omega(n\log n)$, which follows from the classical result of Baur and Strassen [Str73, BS83], and extends the results by Kumar [Kum19], which showed ... more >>>

Mrinal Kumar, Ramprasad Saptharishi, Noam Solomon

A hitting-set generator (HSG) is a polynomial map $Gen:\mathbb{F}^k \to \mathbb{F}^n$ such that for all $n$-variate polynomials $Q$ of small enough circuit size and degree, if $Q$ is non-zero, then $Q\circ Gen$ is non-zero. In this paper, we give a new construction of such a HSG assuming that we have ... more >>>

Mrinal Kumar, Ben Lee Volk

We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various incarnations in computer science; the most significant being in the context of ... more >>>

Chi-Ning Chou, Mrinal Kumar, Noam Solomon

In this note, we give a short, simple and almost completely self contained proof of a classical result of Kaltofen [Kal86, Kal87, Kal89] which shows that if an n variate degree $d$ polynomial f can be computed by an arithmetic circuit of size s, then each of its factors can ... more >>>

Mrinal Kumar, Rafael Mendes de Oliveira, Ramprasad Saptharishi

We show that any $n$-variate polynomial computable by a syntactically multilinear circuit of size $\mathop{poly}(n)$ can be computed by a depth-$4$ syntactically multilinear ($\Sigma\Pi\Sigma\Pi$) circuit of size at most $\exp\left({O\left(\sqrt{n\log n}\right)}\right)$. For degree $d = \omega(n/\log n)$, this improves upon the upper bound of $\exp\left({O(\sqrt{d}\log n)}\right)$ obtained by Tavenas (MFCS ... more >>>

Mrinal Kumar, Ramprasad Saptharishi, Anamay Tengse

The classical lemma of Ore-DeMillo-Lipton-Schwartz-Zippel states that any nonzero polynomial $f(x_1,\ldots, x_n)$ of degree at most $s$ will evaluate to a nonzero value at some point on a grid $S^n \subseteq \mathbb{F}^n$ with $|S| > s$. Thus, there is a deterministic polynomial identity test (PIT) for all degree-$s$ size-$s$ ... more >>>

Abhishek Bhrushundi, Prahladh Harsha, Pooya Hatami, Swastik Kopparty, Mrinal Kumar

In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over $GF(2)= \{0,1\}$. We show the following results for multilinear forms and tensors.

1. Correlation bounds : We show that a random $d$-linear ... more >>>

Mrinal Kumar

We show that over the field of complex numbers, every homogeneous polynomial of degree $d$ can be approximated (in the border complexity sense) by a depth-$3$ arithmetic circuit of top fan-in at most $d+1$. This is quite surprising since there exist homogeneous polynomials $P$ on $n$ variables of degree $2$, ... more >>>

Chi-Ning Chou, Mrinal Kumar, Noam Solomon

In a sequence of fundamental results in the 80's, Kaltofen showed that factors of multivariate polynomials with small arithmetic circuits have small arithmetic circuits. In other words, the complexity class $VP$ is closed under taking factors. A natural question in this context is to understand if other natural classes of ... more >>>

Mrinal Kumar, Ben Lee Volk

We prove a lower bound of $\Omega(n^2/\log^2 n)$ on the size of any syntactically multilinear arithmetic circuit computing some explicit multilinear polynomial $f(x_1, \ldots, x_n)$. Our approach expands and improves upon a result of Raz, Shpilka and Yehudayoff [RSY08], who proved a lower bound of $\Omega(n^{4/3}/\log^2 n)$ for the same ... more >>>

Mrinal Kumar

An algebraic branching program (ABP) is a directed acyclic graph, with a start vertex $s$, and end vertex $t$ and each edge having a weight which is an affine form in $\F[x_1, x_2, \ldots, x_n]$. An ABP computes a polynomial in a natural way, as the sum of weights of ... more >>>

Joshua Grochow, Mrinal Kumar, Michael Saks, Shubhangi Saraf

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 ... more >>>

Mrinal Kumar, Ramprasad Saptharishi

In this paper, we show that there is a family of polynomials $\{P_n\}$, where $P_n$ is a polynomial in $n$ variables of degree at most $d = O(\log^2 n)$, such that

1. $P_n$ can be computed by linear sized homogeneous depth-$5$ circuits.

2. $P_n$ can be computed by ... more >>>

Suryajith Chillara, Mrinal Kumar, Ramprasad Saptharishi, V Vinay

Agrawal and Vinay [AV08] showed how any polynomial size arithmetic circuit can be thought of as a depth four arithmetic circuit of subexponential size. The resulting circuit size in this simulation was more carefully analyzed by Korian [Koiran] and subsequently by Tavenas [Tav13]. We provide a simple proof of this ... more >>>

Michael Forbes, Mrinal Kumar, Ramprasad Saptharishi

We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P \in F[x_1, x_2, \ldots, x_n]$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to {syntactically} computing $P$, when $C \equiv P$ as formal polynomials. In this ... more >>>

Mrinal Kumar, Shubhangi Saraf

In recent years there has been a flurry of activity proving lower bounds for

homogeneous depth-4 arithmetic circuits [GKKS13, FLMS14, KLSS14, KS14c], which has brought us very close to statements that are known to imply VP $\neq$ VNP. It is a big question to go beyond homogeneity, and in ...
more >>>

Mrinal Kumar, Ramprasad Saptharishi

In this paper, we show exponential lower bounds for the class of homogeneous depth-$5$ circuits over all small finite fields. More formally, we show that there is an explicit family $\{P_d : d \in N\}$ of polynomials in $VNP$, where $P_d$ is of degree $d$ in $n = d^{O(1)}$ variables, ... more >>>

Mrinal Kumar, Shubhangi Saraf

We study the complexity of representing polynomials as a sum of products of polynomials in few variables. More precisely, we study representations of the form $$P = \sum_{i = 1}^T \prod_{j = 1}^d Q_{ij}$$

such that each $Q_{ij}$ is an arbitrary polynomial that depends on at most $s$ variables.

Swastik Kopparty, Mrinal Kumar, Michael Saks

We study the problem of indexing irreducible polynomials over finite fields, and give the first efficient algorithm for this problem. Specifically, we show the existence of poly(n, log q)-size circuits that compute a bijection between {1, ... , |S|} and the set S of all irreducible, monic, univariate polynomials of ... more >>>

Mrinal Kumar, Shubhangi Saraf

We prove exponential lower bounds on the size of homogeneous depth 4 arithmetic circuits computing an explicit polynomial in $VP$. Our results hold for the {\it Iterated Matrix Multiplication} polynomial - in particular we show that any homogeneous depth 4 circuit computing the $(1,1)$ entry in the product of $n$ ... more >>>

Mrinal Kumar, Shubhangi Saraf

In this paper, we prove superpolynomial lower bounds for the class of homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP of degree $n$ in $n^2$ variables such that any homogeneous depth 4 arithmetic circuit computing it must have size $n^{\Omega(\log \log n)}$.

Our results extend ... more >>>

Mrinal Kumar, Shubhangi Saraf

In recent years, a very exciting and promising method for proving lower bounds for arithmetic circuits has been proposed. This method combines the method of {\it depth reduction} developed in the works of Agrawal-Vinay [AV08], Koiran [Koi12] and Tavenas [Tav13], and the use of the shifted partial derivative complexity measure ... more >>>

Mrinal Kumar, Shubhangi Saraf

We study the class of homogenous $\Sigma\Pi\Sigma\Pi(r)$ circuits, which are depth 4 homogenous circuits with top fanin bounded by $r$. We show that any homogenous $\Sigma\Pi\Sigma\Pi(r)$ circuit computing the permanent of an $n\times n$ matrix must have size at least $\exp\left(n^{\Omega(1/r)}\right)$.

In a recent result, Gupta, Kamath, Kayal and ... more >>>

Mrinal Kumar, Gaurav Maheshwari, Jayalal Sarma

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 :

$\bullet$ As ... more >>>