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 ... more >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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.

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>

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 >>>