All reports by Author Amir Shpilka:

__
TR22-125
| 9th September 2022
__

Shir Peleg, Amir Shpilka, Ben Lee Volk#### Tensor Reconstruction Beyond Constant Rank

__
TR22-106
| 21st July 2022
__

Suryajith Chillara, Coral Grichener, Amir Shpilka#### On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts

__
TR21-077
| 6th June 2021
__

Shir Peleg, Amir Shpilka, Ben Lee Volk#### Lower Bounds on Stabilizer Rank

__
TR21-014
| 15th February 2021
__

Dori Medini, Amir Shpilka#### Hitting Sets and Reconstruction for Dense Orbits in VP$_e$ and $\Sigma\Pi\Sigma$ Circuits

__
TR17-163
| 2nd November 2017
__

Michael Forbes, Amir Shpilka#### A PSPACE Construction of a Hitting Set for the Closure of Small Algebraic Circuits

__
TR17-007
| 19th January 2017
__

Michael Forbes, Amir Shpilka, Ben Lee Volk#### Succinct Hitting Sets and Barriers to Proving Algebraic Circuits Lower Bounds

Revisions: 1

__
TR16-098
| 16th June 2016
__

Michael Forbes, Amir Shpilka, Iddo Tzameret, Avi Wigderson#### Proof Complexity Lower Bounds from Algebraic Circuit Complexity

__
TR15-184
| 21st November 2015
__

Matthew Anderson, Michael Forbes, Ramprasad Saptharishi, Amir Shpilka, Ben Lee Volk#### Identity Testing and Lower Bounds for Read-$k$ Oblivious Algebraic Branching Programs

__
TR15-025
| 22nd February 2015
__

Shay Moran, Amir Shpilka, Avi Wigderson, Amir Yehudayoff#### Teaching and compressing for low VC-dimension

__
TR14-157
| 27th November 2014
__

Rafael Mendes de Oliveira, Amir Shpilka, Ben Lee Volk#### Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

__
TR14-046
| 8th April 2014
__

Gillat Kol, Shay Moran, Amir Shpilka, Amir Yehudayoff#### Approximate Nonnegative Rank is Equivalent to the Smooth Rectangle Bound

__
TR14-003
| 10th January 2014
__

Zeev Dvir, Rafael Mendes de Oliveira, Amir Shpilka#### Testing Equivalence of Polynomials under Shifts

Revisions: 2
,
Comments: 1

__
TR14-001
| 4th January 2014
__

Swastik Kopparty, Shubhangi Saraf, Amir Shpilka#### Equivalence of Polynomial Identity Testing and Deterministic Multivariate Polynomial Factorization

__
TR13-132
| 23rd September 2013
__

Michael Forbes, Ramprasad Saptharishi, Amir Shpilka#### Pseudorandomness for Multilinear Read-Once Algebraic Branching Programs, in any Order

__
TR13-079
| 2nd June 2013
__

Gillat Kol, Shay Moran, Amir Shpilka, Amir Yehudayoff#### Direct Sum Fails for Zero Error Average Communication

__
TR13-049
| 1st April 2013
__

Amir Shpilka, Ben Lee Volk, Avishay Tal#### On the Structure of Boolean Functions with Small Spectral Norm

Revisions: 1

__
TR13-033
| 1st March 2013
__

Michael Forbes, Amir Shpilka#### Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing

Revisions: 1

__
TR12-115
| 11th September 2012
__

Michael Forbes, Amir Shpilka#### Quasipolynomial-time Identity Testing of Non-Commutative and Read-Once Oblivious Algebraic Branching Programs

Revisions: 1

__
TR11-147
| 2nd November 2011
__

Michael Forbes, Amir Shpilka#### On Identity Testing of Tensors, Low-rank Recovery and Compressed Sensing

__
TR11-079
| 9th May 2011
__

Eli Ben-Sasson, Elena Grigorescu, Ghid Maatouk, Amir Shpilka, Madhu Sudan#### On Sums of Locally Testable Affine Invariant Properties

__
TR11-067
| 25th April 2011
__

Noga Alon, Amir Shpilka, Chris Umans#### On Sunflowers and Matrix Multiplication

Comments: 1

__
TR11-059
| 15th April 2011
__

Elad Haramaty, Amir Shpilka, Madhu Sudan#### Optimal testing of multivariate polynomials over small prime fields

__
TR11-054
| 13th April 2011
__

Arnab Bhattacharyya, Zeev Dvir, Shubhangi Saraf, Amir Shpilka#### Tight lower bounds for 2-query LCCs over finite fields

__
TR11-002
| 9th January 2011
__

Gil Cohen, Amir Shpilka, Avishay Tal#### On the Degree of Univariate Polynomials Over the Integers

__
TR10-199
| 14th December 2010
__

Eli Ben-Sasson, Ghid Maatouk, Amir Shpilka, Madhu Sudan#### Symmetric LDPC codes are not necessarily locally testable

__
TR10-178
| 17th November 2010
__

Amir Shpilka, Avishay Tal#### On the Minimal Fourier Degree of Symmetric Boolean Functions

__
TR10-039
| 10th March 2010
__

Gil Cohen, Amir Shpilka#### On the degree of symmetric functions on the Boolean cube

Comments: 1

__
TR10-036
| 8th March 2010
__

Amir Shpilka, Ilya Volkovich#### On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors

__
TR10-033
| 6th March 2010
__

Shachar Lovett, Partha Mukhopadhyay, Amir Shpilka#### Pseudorandom generators for $\mathrm{CC}_0[p]$ and the Fourier spectrum of low-degree polynomials over finite fields

__
TR10-011
| 22nd January 2010
__

Amir Shpilka, Ilya Volkovich#### Read-Once Polynomial Identity Testing

__
TR09-121
| 22nd November 2009
__

Zohar Karnin, Yuval Rabani, Amir Shpilka#### Explicit Dimension Reduction and Its Applications

__
TR09-116
| 15th November 2009
__

Zohar Karnin, Partha Mukhopadhyay, Amir Shpilka, Ilya Volkovich#### Deterministic identity testing of depth 4 multilinear circuits with bounded top fan-in

__
TR09-080
| 19th September 2009
__

Elad Haramaty, Amir Shpilka#### On the Structure of Cubic and Quartic Polynomials

Revisions: 1

__
TR09-048
| 29th May 2009
__

Parikshit Gopalan, Shachar Lovett, Amir Shpilka#### On the Complexity of Boolean Functions in Different Characteristics

__
TR08-004
| 2nd January 2008
__

Zeev Dvir, Amir Shpilka#### Noisy Interpolating Sets for Low Degree Polynomials

__
TR07-125
| 11th October 2007
__

Ali Juma, Valentine Kabanets, Charles Rackoff, Amir Shpilka#### The black-box query complexity of polynomial summation

__
TR07-122
| 22nd November 2007
__

Zeev Dvir, Amir Shpilka#### Towards Dimension Expanders Over Finite Fields

__
TR07-121
| 21st November 2007
__

Zeev Dvir, Amir Shpilka, Amir Yehudayoff#### Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits

__
TR07-042
| 7th May 2007
__

Zohar Karnin, Amir Shpilka#### Black Box Polynomial Identity Testing of Depth-3 Arithmetic Circuits with Bounded Top Fan-in

Revisions: 2
,
Comments: 1

__
TR06-060
| 4th May 2006
__

Ran Raz, Amir Shpilka, Amir Yehudayoff#### A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits

__
TR05-155
| 10th December 2005
__

Amir Shpilka#### Constructions of low-degree and error-correcting epsilon-biased sets

__
TR05-125
| 2nd November 2005
__

Sofya Raskhodnikova, Dana Ron, Ronitt Rubinfeld, Amir Shpilka, Adam Smith#### Sublinear Algorithms for Approximating String Compressibility and the Distribution Support Size

__
TR05-067
| 28th June 2005
__

Zeev Dvir, Amir Shpilka#### An Improved Analysis of Mergers

__
TR05-044
| 6th April 2005
__

Zeev Dvir, Amir Shpilka#### Locally Decodable Codes with 2 queries and Polynomial Identity Testing for depth 3 circuits

__
TR03-043
| 14th May 2003
__

Elchanan Mossel, Amir Shpilka, Luca Trevisan#### On epsilon-Biased Generators in NC0

__
TR01-060
| 23rd August 2001
__

Amir Shpilka#### Lower bounds for matrix product

__
TR01-035
| 15th April 2001
__

Amir Shpilka#### Affine Projections of Symmetric Polynomials

__
TR00-029
| 30th April 2000
__

Ran Raz, Amir Shpilka#### Lower Bounds for Matrix Product, in Bounded Depth Circuits with Arbitrary Gates

Revisions: 1

__
TR99-023
| 16th June 1999
__

Amir Shpilka, Avi Wigderson#### Depth-3 Arithmetic Formulae over Fields of Characteristic Zero

Shir Peleg, Amir Shpilka, Ben Lee Volk

We give reconstruction algorithms for subclasses of depth-$3$ arithmetic circuits. In particular, we obtain the first efficient algorithm for finding tensor rank, and an optimal tensor decomposition as a sum of rank-one tensors, when given black-box access to a tensor of super-constant rank. Specifically, we obtain the following results:

1. ... more >>>

Suryajith Chillara, Coral Grichener, Amir Shpilka

We say that two given polynomials $f, g \in R[x_1, \ldots, x_n]$, over a ring $R$, are equivalent under shifts if there exists a vector $(a_1, \ldots, a_n)\in R^n$ such that $f(x_1+a_1, \ldots, x_n+a_n) = g(x_1, \ldots, x_n)$. This is a special variant of the polynomial projection problem in Algebraic ... more >>>

Shir Peleg, Amir Shpilka, Ben Lee Volk

The stabilizer rank of a quantum state $\psi$ is the minimal $r$ such that $\left| \psi \right \rangle = \sum_{j=1}^r c_j \left|\varphi_j \right\rangle$ for $c_j \in \mathbb{C}$ and stabilizer states $\varphi_j$. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the ... more >>>

Dori Medini, Amir Shpilka

In this paper we study polynomials in VP$_e$ (polynomial-sized formulas) and in $\Sigma\Pi\Sigma$ (polynomial-size depth-$3$ circuits) whose orbits, under the action of the affine group GL$^{aff}_n({\mathbb F})$, are dense in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms.

As ... more >>>

Michael Forbes, Amir Shpilka

In this paper we study the complexity of constructing a hitting set for $\overline{VP}$, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given ... more >>>

Michael Forbes, Amir Shpilka, Ben Lee Volk

We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike the boolean setting, there has been ... more >>>

Michael Forbes, Amir Shpilka, Iddo Tzameret, Avi Wigderson

We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the ...
more >>>

Matthew Anderson, Michael Forbes, Ramprasad Saptharishi, Amir Shpilka, Ben Lee Volk

Read-$k$ oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs).

In this work, we give an exponential lower bound of $\exp(n/k^{O(k)})$ on the width of any read-$k$ oblivious ABP computing some explicit multilinear polynomial $f$ that is computed by a ...
more >>>

Shay Moran, Amir Shpilka, Avi Wigderson, Amir Yehudayoff

In this work we study the quantitative relation between VC-dimension and two other basic parameters related to learning and teaching. We present relatively efficient constructions of {\em sample compression schemes} and

for classes of low VC-dimension. Let $C$ be a finite boolean concept class of VC-dimension $d$. Set $k ...
more >>>

Rafael Mendes de Oliveira, Amir Shpilka, Ben Lee Volk

In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds we obtain lower bounds for these models.

For depth-3 multilinear formulas, of size $\exp(n^\delta)$, we give a hitting set of size $\exp(\tilde{O}(n^{2/3 + 2\delta/3}))$. ... more >>>

Gillat Kol, Shay Moran, Amir Shpilka, Amir Yehudayoff

We consider two known lower bounds on randomized communication complexity: The smooth rectangle bound and the logarithm of the approximate non-negative rank. Our main result is that they are the same up to a multiplicative constant and a small additive term.

The logarithm of the nonnegative rank is known to ...
more >>>

Zeev Dvir, Rafael Mendes de Oliveira, Amir Shpilka

Two polynomials $f, g \in F[x_1, \ldots, x_n]$ are called shift-equivalent if there exists a vector $(a_1, \ldots, a_n) \in {F}^n$ such that the polynomial identity $f(x_1+a_1, \ldots, x_n+a_n) \equiv g(x_1,\ldots,x_n)$ holds. Our main result is a new randomized algorithm that tests whether two given polynomials are shift equivalent. Our ... 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 >>>

Michael Forbes, Ramprasad Saptharishi, Amir Shpilka

We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in n^(lg^2 n) time. Further, our algorithm is oblivious to the order of the variables. This is the first sub-exponential time algorithm for this model. Furthermore, our result has no known analogue in the ... more >>>

Gillat Kol, Shay Moran, Amir Shpilka, Amir Yehudayoff

We show that in the model of zero error communication complexity, direct sum fails for average communication complexity as well as for external information cost. Our example also refutes a version of a conjecture by Braverman et al. that in the zero error case amortized communication complexity equals external information ... more >>>

Amir Shpilka, Ben Lee Volk, Avishay Tal

In this paper we prove results regarding Boolean functions with small spectral norm (the spectral norm of $f$ is $\|\hat{f}\|_1=\sum_{\alpha}|\hat{f}(\alpha)|$). Specifically, we prove the following results for functions $f:\{0,1\}^n\to \{0,1\}$ with $\|\hat{f}\|_1=A$.

1. There is a subspace $V$ of co-dimension at most $A^2$ such that $f|_V$ is constant.

2. ... more >>>

Michael Forbes, Amir Shpilka

Mulmuley recently gave an explicit version of Noether's Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In ... more >>>

Michael Forbes, Amir Shpilka

We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing (PIT) algorithms for read-once oblivious algebraic branching programs (ABPs). This class has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka), but prior to this work had no known such black-box algorithm. Here we ... more >>>

Michael Forbes, Amir Shpilka

We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms for depth-3 set-multilinear circuits (over arbitrary fields). This class of circuits has an efficient, deterministic, white-box polynomial identity testing algorithm (due to Raz and Shpilka), but has no known such black-box algorithm. We recast this problem as ... more >>>

Eli Ben-Sasson, Elena Grigorescu, Ghid Maatouk, Amir Shpilka, Madhu Sudan

Affine-invariant properties are an abstract class of properties that generalize some

central algebraic ones, such as linearity and low-degree-ness, that have been

studied extensively in the context of property testing. Affine invariant properties

consider functions mapping a big field $\mathbb{F}_{q^n}$ to the subfield $\mathbb{F}_q$ and include all

properties that form ...
more >>>

Noga Alon, Amir Shpilka, Chris Umans

We present several variants of the sunflower conjecture of Erd\H{o}s and Rado and discuss the relations among them.

We then show that two of these conjectures (if true) imply negative answers to questions of Coppersmith and Winograd and Cohn et al. regarding possible approaches for obtaining fast matrix multiplication algorithms. ... more >>>

Elad Haramaty, Amir Shpilka, Madhu Sudan

We consider the problem of testing if a given function $f : \F_q^n \rightarrow \F_q$ is close to a $n$-variate degree $d$ polynomial over the finite field $\F_q$ of $q$ elements. The natural, low-query, test for this property would be to pick the smallest dimension $t = t_{q,d}\approx d/q$ such ... 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 ...
more >>>

Gil Cohen, Amir Shpilka, Avishay Tal

We study the following problem raised by von zur Gathen and Roche:

What is the minimal degree of a nonconstant polynomial $f:\{0,\ldots,n\}\to\{0,\ldots,m\}$?

Clearly, when $m=n$ the function $f(x)=x$ has degree $1$. We prove that when $m=n-1$ (i.e. the point $\{n\}$ is not in the range), it must be the case ... more >>>

Eli Ben-Sasson, Ghid Maatouk, Amir Shpilka, Madhu Sudan

Locally testable codes, i.e., codes where membership in the code is testable with a constant number of queries, have played a central role in complexity theory. It is well known that a code must be a "low-density parity check" (LDPC) code for it to be locally testable, but few LDPC ... more >>>

Amir Shpilka, Avishay Tal

In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function.

Specifically, we prove that for every non-linear and symmetric $f:\{0,1\}^{k} \to \{0,1\}$ there exists a set $\emptyset\neq S\subset[k]$ such that $|S|=O(\Gamma(k)+\sqrt{k})$, and $\hat{f}(S) \neq 0$, where ...
more >>>

Gil Cohen, Amir Shpilka

In this paper we study the degree of non-constant symmetric functions $f:\{0,1\}^n \to \{0,1,\ldots,c\}$, where $c\in

\mathbb{N}$, when represented as polynomials over the real numbers. We show that as long as $c < n$ it holds that deg$(f)=\Omega(n)$. As we can have deg$(f)=1$ when $c=n$, our

result shows a surprising ...
more >>>

Amir Shpilka, Ilya Volkovich

We say that a polynomial $f(x_1,\ldots,x_n)$ is {\em indecomposable} if it cannot be written as a product of two polynomials that are defined over disjoint sets of variables. The {\em polynomial decomposition} problem is defined to be the task of finding the indecomposable factors of a given polynomial. Note that ... more >>>

Shachar Lovett, Partha Mukhopadhyay, Amir Shpilka

In this paper we give the first construction of a pseudorandom generator, with seed length $O(\log n)$, for $\mathrm{CC}_0[p]$, the class of constant-depth circuits with unbounded fan-in $\mathrm{MOD}_p$ gates, for some prime $p$. More accurately, the seed length of our generator is $O(\log n)$ for any constant error $\epsilon>0$. In ... more >>>

Amir Shpilka, Ilya Volkovich

An \emph{arithmetic read-once formula} (ROF for short) is a

formula (a circuit whose underlying graph is a tree) in which the

operations are $\{+,\times\}$ and such that every input variable

labels at most one leaf. A \emph{preprocessed ROF} (PROF for

short) is a ROF in which we are allowed to ...
more >>>

Zohar Karnin, Yuval Rabani, Amir Shpilka

We construct a small set of explicit linear transformations mapping $R^n$ to $R^{O(\log n)}$, such that the $L_2$ norm of

any vector in $R^n$ is distorted by at most $1\pm o(1)$ in at

least a fraction of $1 - o(1)$ of the transformations in the set.

Albeit the tradeoff between ...
more >>>

Zohar Karnin, Partha Mukhopadhyay, Amir Shpilka, Ilya Volkovich

We give the first sub-exponential time deterministic polynomial

identity testing algorithm for depth-$4$ multilinear circuits with

a small top fan-in. More accurately, our algorithm works for

depth-$4$ circuits with a plus gate at the top (also known as

$\Spsp$ circuits) and has a running time of

$\exp(\poly(\log(n),\log(s),k))$ where $n$ is ...
more >>>

Elad Haramaty, Amir Shpilka

In this paper we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results. 1. We give a canonical representation for degree three or four polynomials that have a significant bias ... more >>>

Parikshit Gopalan, Shachar Lovett, Amir Shpilka

Every Boolean function on $n$ variables can be expressed as a unique multivariate polynomial modulo $p$ for every prime $p$. In this work, we study how the degree of a function in one characteristic affects its complexity in other characteristics. We establish the following general principle: functions with low degree ... more >>>

Zeev Dvir, Amir Shpilka

A Noisy Interpolating Set (NIS) for degree $d$ polynomials is a

set $S \subseteq \F^n$, where $\F$ is a finite field, such that

any degree $d$ polynomial $q \in \F[x_1,\ldots,x_n]$ can be

efficiently interpolated from its values on $S$, even if an

adversary corrupts a constant fraction of the values. ...
more >>>

Ali Juma, Valentine Kabanets, Charles Rackoff, Amir Shpilka

For any given Boolean formula $\phi(x_1,\dots,x_n)$, one can

efficiently construct (using \emph{arithmetization}) a low-degree

polynomial $p(x_1,\dots,x_n)$ that agrees with $\phi$ over all

points in the Boolean cube $\{0,1\}^n$; the constructed polynomial

$p$ can be interpreted as a polynomial over an arbitrary field

$\mathbb{F}$. The problem ...
more >>>

Zeev Dvir, Amir Shpilka

In this paper we study the problem of explicitly constructing a

{\em dimension expander} raised by \cite{BISW}: Let $\mathbb{F}^n$

be the $n$ dimensional linear space over the field $\mathbb{F}$.

Find a small (ideally constant) set of linear transformations from

$\F^n$ to itself $\{A_i\}_{i \in I}$ such that for every linear

more >>>

Zeev Dvir, Amir Shpilka, Amir Yehudayoff

In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x_1,...,x_m) that cannot be computed by a depth d arithmetic circuit of small size then there exists ... more >>>

Zohar Karnin, Amir Shpilka

In this paper we consider the problem of determining whether an

unknown arithmetic circuit, for which we have oracle access,

computes the identically zero polynomial. Our focus is on depth-3

circuits with a bounded top fan-in. We obtain the following

results.

1. A quasi-polynomial time deterministic black-box identity testing algorithm ... more >>>

Ran Raz, Amir Shpilka, Amir Yehudayoff

We construct an explicit polynomial $f(x_1,...,x_n)$, with

coefficients in ${0,1}$, such that the size of any syntactically

multilinear arithmetic circuit computing $f$ is at least

$\Omega( n^{4/3} / log^2(n) )$. The lower bound holds over any field.

Amir Shpilka

In this work we give two new constructions of $\epsilon$-biased

generators. Our first construction answers an open question of

Dodis and Smith, and our second construction

significantly extends a result of Mossel et al.

In particular we obtain the following results:

1. We construct a family of asymptotically good binary ... more >>>

Sofya Raskhodnikova, Dana Ron, Ronitt Rubinfeld, Amir Shpilka, Adam Smith

We raise the question of approximating compressibility of a string with respect to a fixed compression scheme, in sublinear time. We study this question in detail for two popular lossless compression schemes: run-length encoding (RLE) and Lempel-Ziv (LZ), and present algorithms and lower bounds for approximating compressibility with respect to ... more >>>

Zeev Dvir, Amir Shpilka

Mergers are functions that transform k (possibly dependent) random sources into a single random source, in a way that ensures that if one of the input sources has min-entropy rate $\delta$ then the output has min-entropy rate close to $\delta$. Mergers have proven to be a very useful tool in ... more >>>

Zeev Dvir, Amir Shpilka

In this work we study two seemingly unrelated notions. Locally Decodable Codes(LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial Identity Testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing ... more >>>

Elchanan Mossel, Amir Shpilka, Luca Trevisan

Cryan and Miltersen recently considered the question

of whether there can be a pseudorandom generator in

NC0, that is, a pseudorandom generator such that every

bit of the output depends on a constant number k of bits

of the seed. They show that for k=3 there is always a

distinguisher; ...
more >>>

Amir Shpilka

We prove lower bounds on the number of product gates in bilinear

and quadratic circuits that

compute the product of two $n \times n$ matrices over finite fields.

In particular we obtain the following results:

1. We show that the number of product gates in any bilinear

(or quadratic) ...
more >>>

Amir Shpilka

In this paper we introduce a new model for computing=20

polynomials - a depth 2 circuit with a symmetric gate at the top=20

and plus gates at the bottom, i.e the circuit computes a=20

symmetric function in linear functions -

$S_{m}^{d}(\ell_1,\ell_2,...,\ell_m)$ ($S_{m}^{d}$ is the $d$'th=20

elementary symmetric polynomial in $m$ ...
more >>>

Ran Raz, Amir Shpilka

We prove super-linear lower bounds for the number of edges

in constant depth circuits with $n$ inputs and up to $n$ outputs.

Our lower bounds are proved for all types of constant depth

circuits, e.g., constant depth arithmetic circuits, constant depth

threshold circuits ...
more >>>

Amir Shpilka, Avi Wigderson

In this paper we prove near quadratic lower bounds for

depth-3 arithmetic formulae 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 ...
more >>>