All reports by Author Shubhangi Saraf:

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TR23-212
| 26th December 2023
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Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, Amir Shpilka#### Exponential Lower Bounds Against Sums of ROABPs

Revisions: 2

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TR23-160
| 29th October 2023
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Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf#### Simple Constructions of Unique Neighbor Expanders from Error-correcting Codes

Revisions: 1

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TR23-032
| 24th March 2023
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Vishwas Bhargava, Shubhangi Saraf, Ilya Volkovich#### Linear Independence, Alternants and Applications

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TR23-017
| 21st February 2023
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Deepanshu Kush, Shubhangi Saraf#### Near-Optimal Set-Multilinear Formula Lower Bounds

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TR22-064
| 2nd May 2022
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Deepanshu Kush, Shubhangi Saraf#### Improved Low-Depth Set-Multilinear Circuit Lower Bounds

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TR21-045
| 22nd March 2021
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Vishwas Bhargava, Shubhangi Saraf, Ilya Volkovich#### Reconstruction Algorithms for Low-Rank Tensors and Depth-3 Multilinear Circuits

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TR20-083
| 30th May 2020
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Eli Ben-Sasson, Dan Carmon, Yuval Ishai, Swastik Kopparty, Shubhangi Saraf#### Proximity Gaps for Reed-Solomon Codes

Revisions: 3

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TR19-104
| 6th August 2019
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Vishwas Bhargava, Shubhangi Saraf, Ilya Volkovich#### Reconstruction of Depth-$4$ Multilinear Circuits

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TR19-080
| 1st June 2019
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Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, Shashwat Silas#### On List Recovery of High-Rate Tensor Codes

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TR19-044
| 28th March 2019
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Eli Ben-Sasson, Lior Goldberg, Swastik Kopparty, Shubhangi Saraf#### DEEP-FRI: Sampling Outside the Box Improves Soundness

Revisions: 2

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TR18-130
| 16th July 2018
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Vishwas Bhargava, Shubhangi Saraf, Ilya Volkovich#### Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree

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TR18-091
| 4th May 2018
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Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, Mary Wootters#### Improved decoding of Folded Reed-Solomon and Multiplicity Codes

Revisions: 2

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TR18-090
| 4th May 2018
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Eli Ben-Sasson, Swastik Kopparty, Shubhangi Saraf#### Worst-case to average case reductions for the distance to a code

Revisions: 1

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TR17-126
| 7th August 2017
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Swastik Kopparty, Shubhangi Saraf#### Local Testing and Decoding of High-Rate Error-Correcting Codes

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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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TR16-122
| 11th August 2016
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Sivakanth Gopi, Swastik Kopparty, Rafael Mendes de Oliveira, Noga Ron-Zewi, Shubhangi Saraf#### Locally testable and Locally correctable Codes Approaching the Gilbert-Varshamov Bound

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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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TR15-110
| 8th July 2015
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Swastik Kopparty, Or Meir, Noga Ron-Zewi, Shubhangi Saraf#### High-rate Locally-testable Codes with Quasi-polylogarithmic Query Complexity

Revisions: 1

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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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TR15-068
| 21st April 2015
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Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf#### High rate locally-correctable and locally-testable codes with sub-polynomial query complexity

Revisions: 3

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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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TR14-026
| 27th February 2014
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Jop Briet, Zeev Dvir, Guangda Hu, Shubhangi Saraf#### Lower Bounds for Approximate LDCs

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TR14-001
| 4th January 2014
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Swastik Kopparty, Shubhangi Saraf, Amir Shpilka#### Equivalence of Polynomial Identity Testing and Deterministic Multivariate Polynomial Factorization

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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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TR13-160
| 20th November 2013
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Zeev Dvir, Shubhangi Saraf, Avi Wigderson#### Breaking the quadratic barrier for 3-LCCs over the Reals

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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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TR12-148
| 7th November 2012
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Eli Ben-Sasson, Ariel Gabizon, Yohay Kaplan, Swastik Kopparty, Shubhangi Saraf#### A new family of locally correctable codes based on degree-lifted algebraic geometry codes

Revisions: 1

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TR12-139
| 2nd November 2012
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Albert Ai, Zeev Dvir, Shubhangi Saraf, Avi Wigderson#### Sylvester-Gallai type theorems for approximate collinearity

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TR12-138
| 2nd November 2012
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Zeev Dvir, Shubhangi Saraf, Avi Wigderson#### Improved rank bounds for design matrices and a new proof of Kelly's theorem

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TR11-054
| 13th April 2011
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Arnab Bhattacharyya, Zeev Dvir, Shubhangi Saraf, Amir Shpilka#### Tight lower bounds for 2-query LCCs over finite fields

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TR11-046
| 2nd April 2011
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Shubhangi Saraf, Ilya Volkovich#### Black-Box Identity Testing of Depth-4 Multilinear Circuits

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TR11-044
| 25th March 2011
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Shubhangi Saraf, Sergey Yekhanin#### Noisy Interpolation of Sparse Polynomials, and Applications

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TR10-148
| 23rd September 2010
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Swastik Kopparty, Shubhangi Saraf, Sergey Yekhanin#### High-rate codes with sublinear-time decoding

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TR09-115
| 13th November 2009
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Swastik Kopparty, Shubhangi Saraf#### Local list-decoding and testing of random linear codes from high-error

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TR09-032
| 16th April 2009
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Neeraj Kayal, Shubhangi Saraf#### Blackbox Polynomial Identity Testing for Depth 3 Circuits

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TR09-004
| 15th January 2009
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Zeev Dvir, Swastik Kopparty, Shubhangi Saraf, Madhu Sudan#### Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers

Revisions: 2

Prerona Chatterjee, Deepanshu Kush, Shubhangi Saraf, Amir Shpilka

In this paper, we prove the first super-polynomial and, in fact, exponential lower bound for the model of sum of read-once oblivious algebraic branching programs (ROABPs). In particular, we give an explicit polynomial such that any sum of ROABPs

(equivalently, sum of *ordered* set-multilinear branching programs, each with a ...
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Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf

In this note, we give very simple constructions of unique neighbor expander graphs starting from spectral or combinatorial expander graphs of mild expansion. These constructions and their analysis are simple variants of the constructions of LDPC error-correcting codes from expanders, given by

Sipser-Spielman~\cite{SS96} (and Tanner~\cite{Tanner81}), and their analysis. We also ...
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Vishwas Bhargava, Shubhangi Saraf, Ilya Volkovich

We develop a new technique for analyzing linear independence of multivariate polynomials. One of our main technical contributions is a \emph{Small Witness for Linear Independence} (SWLI) lemma which states the following.

If the polynomials $f_1,f_2, \ldots, f_k \in \F[X]$ over $X=\{x_1, \ldots, x_n\}$ are $\F$-linearly independent then there exists ...
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Deepanshu Kush, Shubhangi Saraf

The seminal work of Raz (J. ACM 2013) as well as the recent breakthrough results by Limaye, Srinivasan, and Tavenas (FOCS 2021, STOC 2022) have demonstrated a potential avenue for obtaining lower bounds for general algebraic formulas, via strong enough lower bounds for set-multilinear formulas.

In this paper, we make ... more >>>

Deepanshu Kush, Shubhangi Saraf

In this paper, we prove strengthened lower bounds for constant-depth set-multilinear formulas. More precisely, we show that over any field, there is an explicit polynomial $f$ in VNP defined over $n^2$ variables, and of degree $n$, such that any product-depth $\Delta$ set-multilinear formula computing $f$ has size at least $n^{\Omega ... more >>>

Vishwas Bhargava, Shubhangi Saraf, Ilya Volkovich

We give new and efficient black-box reconstruction algorithms for some classes of depth-$3$ arithmetic circuits. As a consequence, we obtain the first efficient algorithm for computing the tensor rank and for finding the optimal tensor decomposition as a sum of rank-one tensors when then input is a {\it constant-rank} tensor. ... more >>>

Eli Ben-Sasson, Dan Carmon, Yuval Ishai, Swastik Kopparty, Shubhangi Saraf

A collection of sets displays a proximity gap with respect to some property if for every set in the collection, either (i) all members are $\delta$-close to the property in relative Hamming distance or (ii) only a tiny fraction of members are $\delta$-close to the property. In particular, no set ... more >>>

Vishwas Bhargava, Shubhangi Saraf, Ilya Volkovich

We present a deterministic algorithm for reconstructing multilinear $\Sigma\Pi\Sigma\Pi(k)$ circuits, i.e. multilinear depth-$4$ circuits with fan-in $k$ at the top $+$ gate. For any fixed $k$, given black-box access to a polynomial $f \in \mathbb{F}[x_{1},x_{2},\ldots ,x_{n}]$ computable by a multilinear $\Sigma\Pi\Sigma\Pi(k)$ circuit of size $s$, the algorithm runs in time ... more >>>

Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, Shashwat Silas

We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS'17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is {\em approximately} locally list recoverable, as well as globally list recoverable ... more >>>

Eli Ben-Sasson, Lior Goldberg, Swastik Kopparty, Shubhangi Saraf

Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worst-case-to-average-case reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. The previous state of the art by [Ben-Sasson, Kopparty, Saraf, CCC 2018] showed that if some member of an affine space $U$ is $\delta$-far in ... more >>>

Vishwas Bhargava, Shubhangi Saraf, Ilya Volkovich

In this paper we study the problem of deterministic factorization of sparse polynomials. We show that if $f \in \mathbb{F}[x_{1},x_{2},\ldots ,x_{n}]$ is a polynomial with $s$ monomials, with individual degrees of its variables bounded by $d$, then $f$ can be deterministically factored in time $s^{\poly(d) \log n}$. Prior to our ... more >>>

Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, Mary Wootters

In this work, we show new and improved error-correcting properties of folded Reed-Solomon codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory. Folded Reed-Solomon codes were the first explicit constructions ... more >>>

Eli Ben-Sasson, Swastik Kopparty, Shubhangi Saraf

Algebraic proof systems reduce computational problems to problems about estimating the distance of a sequence of functions $u=(u_1,\ldots, u_k)$, given as oracles, from a linear error correcting code $V$. The soundness of such systems relies on methods that act ``locally'' on $u$ and map it to a single function $u^*$ ... more >>>

Swastik Kopparty, Shubhangi Saraf

We survey the state of the art in constructions of locally testable

codes, locally decodable codes and locally correctable codes of high rate.

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 >>>

Sivakanth Gopi, Swastik Kopparty, Rafael Mendes de Oliveira, Noga Ron-Zewi, Shubhangi Saraf

One of the most important open problems in the theory

of error-correcting codes is to determine the

tradeoff between the rate $R$ and minimum distance $\delta$ of a binary

code. The best known tradeoff is the Gilbert-Varshamov bound,

and says that for every $\delta \in (0, 1/2)$, there are ...
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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 ...
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Swastik Kopparty, Or Meir, Noga Ron-Zewi, Shubhangi Saraf

An error correcting code is said to be \emph{locally testable} if

there is a test that checks whether a given string is a codeword,

or rather far from the code, by reading only a small number of symbols

of the string. Locally testable codes (LTCs) are both interesting

in their ...
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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, Noga Ron-Zewi, Shubhangi Saraf

In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length $n$, constant rate (which can even be taken arbitrarily close to 1), constant ... 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 >>>

Jop Briet, Zeev Dvir, Guangda Hu, Shubhangi Saraf

We study an approximate version of $q$-query LDCs (Locally Decodable Codes) over the real numbers and prove lower bounds on the encoding length of such codes. A $q$-query $(\alpha,\delta)$-approximate LDC is a set $V$ of $n$ points in $\mathbb{R}^d$ so that, for each $i \in [d]$ there are $\Omega(\delta n)$ ... more >>>

Swastik Kopparty, Shubhangi Saraf, Amir Shpilka

In this paper we show that the problem of deterministically factoring multivariate polynomials reduces to the problem of deterministic polynomial identity testing. Specifically, we show that given an arithmetic circuit (either explicitly or via black-box access) that computes a polynomial $f(X_1,\ldots,X_n)$, the task of computing arithmetic circuits for the factors ... 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 >>>

Zeev Dvir, Shubhangi Saraf, Avi Wigderson

We prove that 3-query linear locally correctable codes over the Reals of dimension $d$ require block length $n>d^{2+\lambda}$ for some fixed, positive $\lambda >0$. Geometrically, this means that if $n$ vectors in $R^d$ are such that each vector is spanned by a linear number of disjoint triples of others, then ... 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 >>>

Eli Ben-Sasson, Ariel Gabizon, Yohay Kaplan, Swastik Kopparty, Shubhangi Saraf

We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry (AG) base-code of block-length $q$ to a lifted-code of block-length $q^m$, for arbitrary integer $m$. The construction generalizes the way degree-$d$, univariate polynomials evaluated over the $q$-element field (also known as Reed-Solomon codes) are "lifted" ... more >>>

Albert Ai, Zeev Dvir, Shubhangi Saraf, Avi Wigderson

We study questions in incidence geometry where the precise position of points is `blurry' (e.g. due to noise, inaccuracy or error). Thus lines are replaced by narrow tubes, and more generally affine subspaces are replaced by their small neighborhood. We show that the presence of a sufficiently large number of ... more >>>

Zeev Dvir, Shubhangi Saraf, Avi Wigderson

We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et. al. (BDWY11) in which they were used to answer questions regarding point configurations. In ... more >>>

Arnab Bhattacharyya, Zeev Dvir, Shubhangi Saraf, Amir Shpilka

A Locally Correctable Code (LCC) is an error correcting code that has a probabilistic

self-correcting algorithm that, with high probability, can correct any coordinate of the

codeword by looking at only a few other coordinates, even if a fraction $\delta$ of the

coordinates are corrupted. LCC's are a stronger form ...
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Shubhangi Saraf, Ilya Volkovich

We study the problem of identity testing for multilinear $\Spsp(k)$ circuits, i.e. multilinear depth-$4$ circuits with fan-in $k$ at the top $+$ gate. We give the first polynomial-time deterministic

identity testing algorithm for such circuits. Our results also hold in the black-box setting.

The running time of our algorithm is ... more >>>

Shubhangi Saraf, Sergey Yekhanin

Let $f\in F_q[x]$ be a polynomial of degree $d\leq q/2.$ It is well-known that $f$ can be uniquely recovered from its values at some $2d$ points even after some small fraction of the values are corrupted. In this paper we establish a similar result for sparse polynomials. We show that ... more >>>

Swastik Kopparty, Shubhangi Saraf, Sergey Yekhanin

Locally decodable codes are error-correcting codes that admit efficient decoding algorithms; any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword. The tradeoff between the rate of a code and the locality/efficiency of its decoding algorithms has been ... more >>>

Swastik Kopparty, Shubhangi Saraf

In this paper, we give surprisingly efficient algorithms for list-decoding and testing

{\em random} linear codes. Our main result is that random sparse linear codes are locally testable and locally list-decodable

in the {\em high-error} regime with only a {\em constant} number of queries.

More precisely, we show that ...
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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 ... more >>>

Zeev Dvir, Swastik Kopparty, Shubhangi Saraf, Madhu Sudan

We extend the ``method of multiplicities'' to get the following results, of interest in combinatorics and randomness extraction.

\begin{enumerate}

\item We show that every Kakeya set in $\F_q^n$, the $n$-dimensional vector space over the finite field on $q$ elements, must be of size at least $q^n/2^n$. This bound is tight ...
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