arXiv subjects

Zeev Dvir

Publications and source records attributed to Zeev Dvir.

36 records · Page 2Linked to original sources

arXiv2014

On the number of rich lines in truly high dimensional sets

We prove a new upper bound on the number of $r$-rich lines (lines with at least $r$ points) in a `truly' $d$-dimensional configuration of points $v_1,\ldots,v_n \in \mathbb{C}^d$. More formally, we show that, if the number of $r$-rich lines is significantly larger than $n^2/r^d$ then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor $r^d$ can be replaced with a tight $r^{d+1}$. If true, this would generalize the classic Szemerédi-Trotter theorem which gives a bound of $n^2/r^3$ on the number of $r$-rich lines in a planar configuration. This conjecture was shown to hold in $\mathbb{R}^3$ in the seminal work of Guth and Katz \cite{GK10} and was also recently proved over $\mathbb{R}^4$ (under some additional restrictions) \cite{SS14}. For the special case of arithmetic progressions ($r$ collinear points that are evenly distanced) we give a bound that is tight up to low order terms, showing that a $d$-dimensional grid achieves the largest number of $r$-term progressions. The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree $r-2$ Veronese embedding takes $r$-collinear points to $r$ linearly dependent images. Hence, each collinear $r$-tuple of points, gives us a dependent $r$-tuple of images. We then use the design-matrix method of \cite{BDWY12} to convert these 'local' linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set.

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arXiv2014

2-Server PIR with sub-polynomial communication

A 2-server Private Information Retrieval (PIR) scheme allows a user to retrieve the $i$th bit of an $n$-bit database replicated among two servers (which do not communicate) while not revealing any information about $i$ to either server. In this work we construct a 1-round 2-server PIR with total communication cost $n^{O({\sqrt{\log\log n/\log n}})}$. This improves over the currently known 2-server protocols which require $O(n^{1/3})$ communication and matches the communication cost of known 3-server PIR schemes. Our improvement comes from reducing the number of servers in existing protocols, based on Matching Vector Codes, from 3 or 4 servers to 2. This is achieved by viewing these protocols in an algebraic way (using polynomial interpolation) and extending them using partial derivatives.

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arXiv2014

A quantitative variant of the multi-colored Motzkin-Rabin theorem

We prove a quantitative version of the multi-colored Motzkin-Rabin theorem in the spirit of [BDWY12]: Let $V_1,\ldots,V_n \subset R^d$ be $n$ disjoint sets of points (of $n$ `colors'). Suppose that for every $V_i$ and every point $v \in V_i$ there are at least $δ|V_i|$ other points $u \in V_i$ so that the line connecting $v$ and $u$ contains a third point of another color. Then the union of the points in all $n$ sets is contained in a subspace of dimension bounded by a function of $n$ and $δ$ alone.

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arXiv2014

Factors of Sparse Polynomials are Sparse

This paper was removed due to an error in the proof (Claim 4.12 as stated is not true). The authors would like to thank Ilya Volkovich for pointing out a counterexample to this paper's main result in positive characteristic: If $F$ is a field with prime characteristic $p$, then the polynomial $x_1^p + x_2^p + \ldots + x^n^p$ has the following factor: $(x_1+x_2+ \ldots + x_n)^{p-1}$, which has sparsity $n^p$.

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arXiv2014

Lower Bounds for Approximate LDC

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 $(α,δ)$-approximate LDC is a set $V$ of $n$ points in $\mathbb{R}^d$ so that, for each $i \in [d]$ there are $Ω(δn)$ disjoint $q$-tuples $(\vec{u}_1,\ldots,\vec{u}_q) $ in $V$ so that $\text{span}(\vec{u}_1,\ldots,\vec{u}_q)$ contains a unit vector whose $i$'th coordinate is at least $α$. We prove exponential lower bounds of the form $n \geq 2^{Ω(αδ\sqrt{d})}$ for the case $q=2$ and, in some cases, stronger bounds (exponential in $d$).

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arXiv2014

Testing Equivalence of Polynomials under Shifts

Two polynomials $f, g \in \mathbb{F}[x_1, \ldots, x_n]$ are called shift-equivalent if there exists a vector $(a_1, \ldots, a_n) \in \mathbb{F}^n$ 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 algorithm runs in time polynomial in the circuit size of the polynomials, to which it is given black box access. This complements a previous work of Grigoriev (Theoretical Computer Science, 1997) who gave a deterministic algorithm running in time $n^{O(d)}$ for degree $d$ polynomials. Our algorithm uses randomness only to solve instances of the Polynomial Identity Testing (PIT) problem. Hence, if one could de-randomize PIT (a long-standing open problem in complexity) a de-randomization of our algorithm would follow. This establishes an equivalence between de-randomizing shift-equivalence testing and de-randomizing PIT (both in the black-box and the white-box setting). For certain restricted models, such as Read Once Branching Programs, we already obtain a deterministic algorithm using existing PIT results.

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arXiv2014

Affine extractors over large fields with exponential error

We describe a construction of explicit affine extractors over large finite fields with exponentially small error and linear output length. Our construction relies on a deep theorem of Deligne giving tight estimates for exponential sums over smooth varieties in high dimensions.

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arXiv2013

Breaking the quadratic barrier for 3-LCCs over the Reals

We prove that 3-query linear locally correctable codes over the Reals of dimension $d$ require block length $n>d^{2+λ}$ for some fixed, positive $λ>0$. 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 it must be that $n > d^{2+λ}$. This improves the known quadratic lower bounds (e.g. {KdW04, Wood07}). While a modest improvement, we expect that the new techniques introduced in this work will be useful for further progress on lower bounds of locally correctable and decodable codes with more than 2 queries, possibly over other fields as well. Our proof introduces several new ideas to existing lower bound techniques, several of which work over every field. At a high level, our proof has two parts, {\it clustering} and {\it random restriction}. The clustering step uses a powerful theorem of Barthe from convex geometry. It can be used (after preprocessing our LCC to be {\it balanced}), to apply a basis change (and rescaling) of the vectors, so that the resulting unit vectors become {\it nearly isotropic}. This together with the fact that any LCC must have many `correlated' pairs of points, lets us deduce that the vectors must have a surprisingly strong geometric clustering, and hence also combinatorial clustering with respect to the spanning triples. In the restriction step, we devise a new variant of the dimension reduction technique used in previous lower bounds, which is able to take advantage of the combinatorial clustering structure above. The analysis of our random projection method reduces to a simple (weakly) random graph process, and works over any field.

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arXiv2013

Incidence Theorems and Their Applications

We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we will discuss are: (1) Counting incidences: Given a set (or several sets) of geometric objects (lines, points, etc..), what is the maximum number of incidences (or intersections) that can exist between elements in different sets? We will see several results of this type, such as the Szemeredi-Trotter theorem, over the reals and over finite fields and discuss their applications in combinatorics (e.g., in the recent solution of Guth and Katz to Erdos' distance problem) and in computer science (in explicit constructions of multi-source extractors). (2) Kakeya type problems: These problems deal with arrangements of lines that point in different directions. The goal is to try and understand to what extent these lines can overlap one another. We will discuss these questions both over the reals and over finite fields and see how they come up in the theory of randomness-extractors. (3) Sylvester-Gallai type problems: In this type of problems, one is presented with a configuration of points that contain many `local' dependencies (e.g., three points on a line) and is asked to derive a bound on the dimension of the span of all points. We will discuss several recent results of this type, over various fields, and see their connection to the theory of locally correctable error-correcting codes. Throughout the different parts of the survey, two types of techniques will make frequent appearance. One is the polynomial method, which uses polynomial interpolation to impose an algebraic structure on the problem at hand. The other recurrent techniques will come from the area of additive combinatorics.

math.CO↗

arXiv2013

Matching-Vector Families and LDCs Over Large Modulo

We prove new upper bounds on the size of families of vectors in $\Z_m^n$ with restricted modular inner products, when $m$ is a large integer. More formally, if $\vec{u}_1,\ldots,\vec{u}_t \in \Z_m^n$ and $\vec{v}_1,\ldots,\vec{v}_t \in \Z_m^n$ satisfy $\langle\vec{u}_i,\vec{v}_i\rangle\equiv0\pmod m$ and $\langle\vec{u}_i,\vec{v}_j\rangle\not\equiv0\pmod m$ for all $i\neq j\in[t]$, we prove that $t \leq O(m^{n/2+8.47})$. This improves a recent bound of $t \leq m^{n/2 + O(\log(m))}$ by \cite{BDL13} and is the best possible up to the constant 8.47 when $m$ is sufficiently larger than $n$. The maximal size of such families, called `Matching-Vector families', shows up in recent constructions of locally decodable error correcting codes (LDCs) and determines the rate of the code. Using our result we are able to show that these codes, called Matching-Vector codes, must have encoding length at least $K^{19/18}$ for $K$-bit messages, regardless of their query complexity. This improves a known super linear bound of $ K2^{Ω({\sqrt{\log K}})}$ proved in \cite{DGY11}.

math.CO↗

arXiv2013

New Lower Bounds for Matching Vector Codes

A Matching Vector (MV) family modulo $m$ is a pair of ordered lists $U=(u_1,...,u_t)$ and $V=(v_1,...,v_t)$ where $u_i,v_j \in \mathbb{Z}_m^n$ with the following inner product pattern: for any $i$, $< u_i,v_i>=0$, and for any $i \ne j$, $< u_i,v_j> \ne 0$. A MV family is called $q$-restricted if inner products $< u_i,v_j>$ take at most $q$ different values. Our interest in MV families stems from their recent application in the construction of sub-exponential locally decodable codes (LDCs). There, $q$-restricted MV families are used to construct LDCs with $q$ queries, and there is special interest in the regime where $q$ is constant. When $m$ is a prime it is known that such constructions yield codes with exponential block length. However, for composite $m$ the behaviour is dramatically different. A recent work by Efremenko [STOC 2009] (based on an approach initiated by Yekhanin [JACM 2008]) gives the first sub-exponential LDC with constant queries. It is based on a construction of a MV family of super-polynomial size by Grolmusz [Combinatorica 2000] modulo composite $m$. In this work, we prove two lower bounds on the block length of LDCs which are based on black box construction using MV families. When $q$ is constant (or sufficiently small), we prove that such LDCs must have a quadratic block length. When the modulus $m$ is constant (as it is in the construction of Efremenko) we prove a super-polynomial lower bound on the block-length of the LDCs, assuming a well-known conjecture in additive combinatorics, the polynomial Freiman-Ruzsa conjecture over $\mathbb{Z}_m$.

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arXiv2012

Improved rank bounds for design matrices and a new proof of Kelly's theorem

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 this work we derive near-optimal rank bounds for these matrices and use them to obtain asymptotically tight bounds in many of the geometric applications. As a consequence of our improved analysis, we also obtain a new, linear algebraic, proof of Kelly's theorem, which is the complex analog of the Sylvester-Gallai theorem.

math.CO↗

arXiv2012

Sylvester-Gallai type theorems for approximate collinearity

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 approximately collinear triples in a set of points in d dimensional complex space implies that the points are close to a low dimensional affine subspace. This can be viewed as a stable variant of the Sylvester-Gallai theorem and its extensions. Building on the recently found connection between Sylvester-Gallai type theorems and complex Locally Correctable Codes (LCCs), we define the new notion of stable LCCs, in which the (local) correction procedure can also handle small perturbations in the euclidean metric. We prove that such stable codes with constant query complexity do not exist. No impossibility results were known in any such local setting for more than 2 queries.

math.CO↗

arXiv2012

Variety Evasive Sets

We give an explicit construction of a large subset of F^n, where F is a finite field, that has small intersection with any affine variety of fixed dimension and bounded degree. Our construction generalizes a recent result of Dvir and Lovett (STOC 2012) who considered varieties of degree one (affine subspaces).

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arXiv2011

Subspace Evasive Sets

In this work we describe an explicit, simple, construction of large subsets of F^n, where F is a finite field, that have small intersection with every k-dimensional affine subspace. Interest in the explicit construction of such sets, termed subspace-evasive sets, started in the work of Pudlak and Rodl (2004) who showed how such constructions over the binary field can be used to construct explicit Ramsey graphs. More recently, Guruswami (2011) showed that, over large finite fields (of size polynomial in n), subspace evasive sets can be used to obtain explicit list-decodable codes with optimal rate and constant list-size. In this work we construct subspace evasive sets over large fields and use them to reduce the list size of folded Reed-Solomon codes form poly(n) to a constant.

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arXiv2011

Rank Bounds for Design Matrices with Applications to Combinatorial Geometry and Locally Correctable Codes

A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns intersect in at most t rows. We prove that the rank of any (q,k,t)-design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least n - (qtn/2k)^2 . Using this result we derive the following applications: (1) Impossibility results for 2-query LCCs over the complex numbers: A 2-query locally correctable code (LCC) is an error correcting code in which every codeword coordinate can be recovered, probabilistically, by reading at most two other code positions. Such codes have numerous applications and constructions (with exponential encoding length) are known over finite fields of small characteristic. We show that infinite families of such linear 2-query LCCs do not exist over the complex numbers. (2) Generalization of results in combinatorial geometry: We prove a quantitative analog of the Sylvester-Gallai theorem: Let $v_1,...,v_m$ be a set of points in $\C^d$ such that for every $i \in [m]$ there exists at least $δm$ values of $j \in [m]$ such that the line through $v_i,v_j$ contains a third point in the set. We show that the dimension of $\{v_1,...,v_m \}$ is at most $O(1/δ^2)$. Our results generalize to the high dimensional case (replacing lines with planes, etc.) and to the case where the points are colored (as in the Motzkin-Rabin Theorem).

math.CO↗

arXiv2009

Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers

We extend the "method of multiplicities" to get the following results, of interest in combinatorics and randomness extraction. (A) We show that every Kakeya set (a set of points that contains a line in every direction) in $\F_q^n$ must be of size at least $q^n/2^n$. This bound is tight to within a $2 + o(1)$ factor for every $n$ as $q \to \infty$, compared to previous bounds that were off by exponential factors in $n$. (B) We give improved randomness extractors and "randomness mergers". Mergers are seeded functions that take as input $Λ$ (possibly correlated) random variables in $\{0,1\}^N$ and a short random seed and output a single random variable in $\{0,1\}^N$ that is statistically close to having entropy $(1-δ) \cdot N$ when one of the $Λ$ input variables is distributed uniformly. The seed we require is only $(1/δ)\cdot \log Λ$-bits long, which significantly improves upon previous construction of mergers. (C) Using our new mergers, we show how to construct randomness extractors that use logarithmic length seeds while extracting $1 - o(1)$ fraction of the min-entropy of the source. The "method of multiplicities", as used in prior work, analyzed subsets of vector spaces over finite fields by constructing somewhat low degree interpolating polynomials that vanish on every point in the subset {\em with high multiplicity}. The typical use of this method involved showing that the interpolating polynomial also vanished on some points outside the subset, and then used simple bounds on the number of zeroes to complete the analysis. Our augmentation to this technique is that we prove, under appropriate conditions, that the interpolating polynomial vanishes {\em with high multiplicity} outside the set. This novelty leads to significantly tighter analyses.

math.CO↗

arXiv2008

On the size of Kakeya sets in finite fields

A Kakeya set is a subset of F^n, where F is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least C_n * q^n, where C_n depends only on n. This improves the previously best lower bound for general n of ~q^{4n/7}.

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