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Electronic Colloquium on Computational Complexity

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All reports by Author Amir Shpilka:

TR17-007 | 19th January 2017
Michael Forbes, Amir Shpilka, Ben Lee Volk

Succinct Hitting Sets and Barriers to Proving Algebraic Circuits Lower Bounds

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 >>>

TR16-098 | 16th June 2016
Michael Forbes, Amir Shpilka, Iddo Tzameret, Avi Wigderson

Proof Complexity Lower Bounds from Algebraic Circuit Complexity

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 >>>

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

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 >>>

TR15-025 | 22nd February 2015
Shay Moran, Amir Shpilka, Avi Wigderson, Amir Yehudayoff

Teaching and compressing for low VC-dimension

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 >>>

TR14-046 | 8th April 2014
Gillat Kol, Shay Moran, Amir Shpilka, Amir Yehudayoff

Approximate Nonnegative Rank is Equivalent to the Smooth Rectangle Bound

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 >>>

TR14-003 | 10th January 2014
Zeev Dvir, Rafael Mendes de Oliveira, Amir Shpilka

Testing Equivalence of Polynomials under Shifts

Revisions: 2 , Comments: 1

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 >>>

TR14-001 | 4th January 2014
Swastik Kopparty, Shubhangi Saraf, Amir Shpilka

Equivalence of Polynomial Identity Testing and Deterministic Multivariate Polynomial Factorization

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 >>>

TR13-132 | 23rd September 2013
Michael Forbes, Ramprasad Saptharishi, Amir Shpilka

Pseudorandomness for Multilinear Read-Once Algebraic Branching Programs, in any Order

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 >>>

TR13-079 | 2nd June 2013
Gillat Kol, Shay Moran, Amir Shpilka, Amir Yehudayoff

Direct Sum Fails for Zero Error Average Communication

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 >>>

TR13-049 | 1st April 2013
Amir Shpilka, Ben Lee Volk, Avishay Tal

On the Structure of Boolean Functions with Small Spectral Norm

Revisions: 1

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 >>>

TR13-033 | 1st March 2013
Michael Forbes, Amir Shpilka

Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing

Revisions: 1

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 >>>

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

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 >>>

TR11-147 | 2nd November 2011
Michael Forbes, Amir Shpilka

On Identity Testing of Tensors, Low-rank Recovery and Compressed Sensing

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 >>>

TR11-079 | 9th May 2011
Eli Ben-Sasson, Elena Grigorescu, Ghid Maatouk, Amir Shpilka, Madhu Sudan

On Sums of Locally Testable Affine Invariant Properties

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 >>>

TR11-067 | 25th April 2011
Noga Alon, Amir Shpilka, Chris Umans

On Sunflowers and Matrix Multiplication

Comments: 1

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 >>>

TR11-059 | 15th April 2011
Elad Haramaty, Amir Shpilka, Madhu Sudan

Optimal testing of multivariate polynomials over small prime fields

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 >>>

TR11-054 | 13th April 2011
Arnab Bhattacharyya, Zeev Dvir, Shubhangi Saraf, Amir Shpilka

Tight lower bounds for 2-query LCCs over finite fields

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 >>>

TR11-002 | 9th January 2011
Gil Cohen, Amir Shpilka, Avishay Tal

On the Degree of Univariate Polynomials Over the Integers

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 >>>

TR10-199 | 14th December 2010
Eli Ben-Sasson, Ghid Maatouk, Amir Shpilka, Madhu Sudan

Symmetric LDPC codes are not necessarily locally testable

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 >>>

TR10-178 | 17th November 2010
Amir Shpilka, Avishay Tal

On the Minimal Fourier Degree of Symmetric Boolean Functions

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 >>>

TR10-039 | 10th March 2010
Gil Cohen, Amir Shpilka

On the degree of symmetric functions on the Boolean cube

Comments: 1

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 >>>

TR10-036 | 8th March 2010
Amir Shpilka, Ilya Volkovich

On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors

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 >>>

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

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 >>>

TR10-011 | 22nd January 2010
Amir Shpilka, Ilya Volkovich

Read-Once Polynomial Identity Testing

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 >>>

TR09-121 | 22nd November 2009
Zohar Karnin, Yuval Rabani, Amir Shpilka

Explicit Dimension Reduction and Its Applications

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 >>>

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

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 >>>

TR09-080 | 19th September 2009
Elad Haramaty, Amir Shpilka

On the Structure of Cubic and Quartic Polynomials

Revisions: 1

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 >>>

TR09-048 | 29th May 2009
Parikshit Gopalan, Shachar Lovett, Amir Shpilka

On the Complexity of Boolean Functions in Different Characteristics

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 >>>

TR08-004 | 2nd January 2008
Zeev Dvir, Amir Shpilka

Noisy Interpolating Sets for Low Degree Polynomials

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 >>>

TR07-125 | 11th October 2007
Ali Juma, Valentine Kabanets, Charles Rackoff, Amir Shpilka

The black-box query complexity of polynomial summation

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 >>>

TR07-122 | 22nd November 2007
Zeev Dvir, Amir Shpilka

Towards Dimension Expanders Over Finite Fields

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 >>>

TR07-121 | 21st November 2007
Zeev Dvir, Amir Shpilka, Amir Yehudayoff

Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits

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 >>>

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

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

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

TR06-060 | 4th May 2006
Ran Raz, Amir Shpilka, Amir Yehudayoff

A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits

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.

more >>>

TR05-155 | 10th December 2005
Amir Shpilka

Constructions of low-degree and error-correcting epsilon-biased sets

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 >>>

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

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 >>>

TR05-067 | 28th June 2005
Zeev Dvir, Amir Shpilka

An Improved Analysis of Mergers

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 >>>

TR05-044 | 6th April 2005
Zeev Dvir, Amir Shpilka

Locally Decodable Codes with 2 queries and Polynomial Identity Testing for depth 3 circuits

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 >>>

TR03-043 | 14th May 2003
Elchanan Mossel, Amir Shpilka, Luca Trevisan

On epsilon-Biased Generators in NC0

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 >>>

TR01-060 | 23rd August 2001
Amir Shpilka

Lower bounds for matrix product

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 >>>

TR01-035 | 15th April 2001
Amir Shpilka

Affine Projections of Symmetric Polynomials

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 >>>

TR00-029 | 30th April 2000
Ran Raz, Amir Shpilka

Lower Bounds for Matrix Product, in Bounded Depth Circuits with Arbitrary Gates

Revisions: 1

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 >>>

TR99-023 | 16th June 1999
Amir Shpilka, Avi Wigderson

Depth-3 Arithmetic Formulae over Fields of Characteristic Zero

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 >>>

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