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Shachar Lovett

Publications and source records attributed to Shachar Lovett.

88 records · Page 5Linked to original sources
arXiv2012

Testing Low Complexity Affine-Invariant Properties

Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affine-invariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the Reed-Muller code over F_p of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials except that low degree is preserved by composition with affine maps. The complexity of an affine-invariant property P refers to the maximum complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of linear forms used to characterize P. A more precise statement of our main result is that for any fixed prime p >=2 and fixed integer R >= 2, any affine-invariant property P of functions f: F_p^n -> [R] is testable, assuming the complexity of the property is less than p. Our proof involves developing analogs of graph-theoretic techniques in an algebraic setting, using tools from higher-order Fourier analysis.

cs.CC↗
arXiv2012

A Tail Bound for Read-k Families of Functions

We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,...,Y_r$ are arbitrary Boolean functions of independent random variables $X_1,...,X_m$, modulo a restriction that every $X_i$ influences at most $k$ of the variables $Y_1,...,Y_r$.

cs.DM↗
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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arXiv2012

On the Furthest Hyperplane Problem and Maximal Margin Clustering

This paper introduces the Furthest Hyperplane Problem (FHP), which is an unsupervised counterpart of Support Vector Machines. Given a set of n points in Rd, the objective is to produce the hyperplane (passing through the origin) which maximizes the separation margin, that is, the minimal distance between the hyperplane and any input point. To the best of our knowledge, this is the first paper achieving provable results regarding FHP. We provide both lower and upper bounds to this NP-hard problem. First, we give a simple randomized algorithm whose running time is n^O(1/θ^2) where θ is the optimal separation margin. We show that its exponential dependency on 1/θ^2 is tight, up to sub-polynomial factors, assuming SAT cannot be solved in sub-exponential time. Next, we give an efficient approxima- tion algorithm. For any α \in [0, 1], the algorithm produces a hyperplane whose distance from at least 1 - 5α fraction of the points is at least α times the optimal separation margin. Finally, we show that FHP does not admit a PTAS by presenting a gap preserving reduction from a particular version of the PCP theorem.

cs.CC↗
arXiv2011

An additive combinatorics approach to the log-rank conjecture in communication complexity

For a $\{0,1\}$-valued matrix $M$ let $\rm{CC}(M)$ denote the deterministic communication complexity of the boolean function associated with $M$. The log-rank conjecture of Lovász and Saks [FOCS 1988] states that $\rm{CC}(M) \leq \log^c(\rm{rank}(M))$ for some absolute constant $c$ where $\rm{rank}(M)$ denotes the rank of $M$ over the field of real numbers. We show that $\rm{CC}(M)\leq c \cdot \rm{rank}(M)/\log \rm{rank}(M)$ for some absolute constant $c$, assuming a well-known conjecture from additive combinatorics known as the Polynomial Freiman-Ruzsa (PFR) conjecture. Our proof is based on the study of the "approximate duality conjecture" which was recently suggested by Ben-Sasson and Zewi [STOC 2011] and studied there in connection to the PFR conjecture. First we improve the bounds on approximate duality assuming the PFR conjecture. Then we use the approximate duality conjecture (with improved bounds) to get the aforementioned upper bound on the communication complexity of low-rank martices, where this part uses the methodology suggested by Nisan and Wigderson [Combinatorica 1995].

cs.CC↗
arXiv2011

Probabilistic existence of rigid combinatorial structures

We show the existence of rigid combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, $t$-designs, and $t$-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. The proof of existence is probabilistic. We show that a randomly chosen such object has the required properties with positive yet tiny probability. The main technical ingredient is a special local central limit theorem for suitable lattice random walks with finitely many steps.

math.CO↗
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

Higher-order Fourier analysis of $\mathbb{F}_p^n$ and the complexity of systems of linear forms

Consider a subset $A$ of $\mathbb{F}_p^n$ and a decomposition of its indicator function as the sum of two bounded functions $1_A=f_1+f_2$. For every family of linear forms, we find the smallest degree of uniformity $k$ such that assuming that $\|f_2\|_{U^k}$ is sufficiently small, it is possible to discard $f_2$ and replace $1_A$ with $f_1$ in the average over this family of linear forms, affecting it only negligibly. Previously, Gowers and Wolf solved this problem for the case where $f_1$ is a constant function. Furthermore, our main result solves Problem 7.6 in [W. T. Gowers and J. Wolf. Linear forms and higher-degree uniformity for functions on $\mathbb{F}_p^n$. Geom. Funct. Anal., 21(1):36--69, 2011] regarding the analytic averages that involve more than one subset of $\mathbb{F}_p^n$.] regarding the analytic averages that involve more than one subset of $\mathbb{F}_p^n$.

math.NT↗
arXiv2010

Equivalence of polynomial conjectures in additive combinatorics

We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small doubling. The second is the inverse Gowers conjecture for $U^3$, which relates to functions which locally look like quadratics. In both cases a weak form, with exponential decay of parameters is known, and a strong form with only a polynomial loss of parameters is conjectured. Our main result is that the two conjectures are in fact equivalent.

math.CO↗
arXiv2009

Polynomial Threshold Functions: Structure, Approximation and Pseudorandomness

We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows that low-degree polynomial threshold functions cannot approximate any function with many influential variables. We provide a couple of examples where this technique yields tight approximation bounds. Our second result relates to constructing pseudorandom generators fooling low-degree polynomial threshold functions. This problem has received attention recently, where Diakonikolas et al proved that $k$-wise independence suffices to fool linear threshold functions. We prove that any low-degree polynomial threshold function, which can be represented as a function of a small number of linear threshold functions, can also be fooled by $k$-wise independence. We view this as an important step towards fooling general polynomial threshold functions, and we discuss a plausible approach achieving this goal based on our techniques. Our results combine tools from real approximation theory, hyper-contractive inequalities and probabilistic methods. In particular, we develop several new tools in approximation theory which may be of independent interest.

cs.CC↗
arXiv2009

The density of weights of Generalized Reed--Muller codes

We study the density of the weights of Generalized Reed--Muller codes. Let $RM_p(r,m)$ denote the code of multivariate polynomials over $\F_p$ in $m$ variables of total degree at most $r$. We consider the case of fixed degree $r$, when we let the number of variables $m$ tend to infinity. We prove that the set of relative weights of codewords is quite sparse: for every $α\in [0,1]$ which is not rational of the form $\frac{\ell}{p^k}$, there exists an interval around $α$ in which no relative weight exists, for any value of $m$. This line of research is to the best of our knowledge new, and complements the traditional lines of research, which focus on the weight distribution and the divisibility properties of the weights. Equivalently, we study distributions taking values in a finite field, which can be approximated by distributions coming from constant degree polynomials, where we do not bound the number of variables. We give a complete characterization of all such distributions.

cs.IT↗
arXiv2008

The List-Decoding Size of Reed-Muller Codes

In this work we study the list-decoding size of Reed-Muller codes. Given a received word and a distance parameter, we are interested in bounding the size of the list of Reed-Muller codewords that are within that distance from the received word. Previous bounds of Gopalan, Klivans and Zuckerman \cite{GKZ08} on the list size of Reed-Muller codes apply only up to the minimum distance of the code. In this work we provide asymptotic bounds for the list-decoding size of Reed-Muller codes that apply for {\em all} distances. Additionally, we study the weight distribution of Reed-Muller codes. Prior results of Kasami and Tokura \cite{KT70} on the structure of Reed-Muller codewords up to twice the minimum distance, imply bounds on the weight distribution of the code that apply only until twice the minimum distance. We provide accumulative bounds for the weight distribution of Reed-Muller codes that apply to {\em all} distances.

cs.IT↗
arXiv2008

Inverse Conjecture for the Gowers norm is false

Let $p$ be a fixed prime number, and $N$ be a large integer. The 'Inverse Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a function $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant independent of $N$, then $f$ can be non-trivially approximated by a degree $d-1$ polynomial. The conjecture is known to hold for $d=2,3$ and for any prime $p$. In this paper we show the conjecture to be false for $p=2$ and for $d = 4$, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao \cite{gt07}. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel \cite{ab} to show inapproximability of certain functions by low-degree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime $p$, for $d = p^2$.

math.CO↗
arXiv2008

Worst Case to Average Case Reductions for Polynomials

A degree-$d$ polynomial $p$ in $n$ variables over a field $\F$ is {\em equidistributed} if it takes on each of its $|\F|$ values close to equally often, and {\em biased} otherwise. We say that $p$ has a {\em low rank} if it can be expressed as a bounded combination of polynomials of lower degree. Green and Tao [gt07] have shown that bias imply low rank over large fields (i.e. for the case $d < |\F|$). They have also conjectured that bias imply low rank over general fields. In this work we affirmatively answer their conjecture. Using this result we obtain a general worst case to average case reductions for polynomials. That is, we show that a polynomial that can be {\em approximated} by few polynomials of bounded degree, can be {\em computed} by few polynomials of bounded degree. We derive some relations between our results to the construction of pseudorandom generators, and to the question of testing concise representations.

math.CO↗
arXiv2008

Lower bounds for adaptive linearity tests

Linearity tests are randomized algorithms which have oracle access to the truth table of some function f, and are supposed to distinguish between linear functions and functions which are far from linear. Linearity tests were first introduced by (Blum, Luby and Rubenfeld, 1993), and were later used in the PCP theorem, among other applications. The quality of a linearity test is described by its correctness c - the probability it accepts linear functions, its soundness s - the probability it accepts functions far from linear, and its query complexity q - the number of queries it makes. Linearity tests were studied in order to decrease the soundness of linearity tests, while keeping the query complexity small (for one reason, to improve PCP constructions). Samorodnitsky and Trevisan (Samorodnitsky and Trevisan 2000) constructed the Complete Graph Test, and prove that no Hyper Graph Test can perform better than the Complete Graph Test. Later in (Samorodnitsky and Trevisan 2006) they prove, among other results, that no non-adaptive linearity test can perform better than the Complete Graph Test. Their proof uses the algebraic machinery of the Gowers Norm. A result by (Ben-Sasson, Harsha and Raskhodnikova 2005) allows to generalize this lower bound also to adaptive linearity tests. We also prove the same optimal lower bound for adaptive linearity test, but our proof technique is arguably simpler and more direct than the one used in (Samorodnitsky and Trevisan 2006). We also study, like (Samorodnitsky and Trevisan 2006), the behavior of linearity tests on quadratic functions. However, instead of analyzing the Gowers Norm of certain functions, we provide a more direct combinatorial proof, studying the behavior of linearity tests on random quadratic functions...

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