arXiv subjects

Nathan Keller

Publications and source records attributed to Nathan Keller.

36 records · Page 2Linked to original sources

arXiv2018

On the structure of subsets of the discrete cube with small edge boundary

The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of an $m$-element subset of $\{0,1\}^{n}$; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on $\{0,1\}^n$. We show that for any $m$-element subset $\mathcal{F} \subset \{0,1\}^n$ and any integer $l$, if the edge boundary of $\mathcal{F}$ has size at most $g_n(m)+l$, then there exists an extremal family $\mathcal{G} \subset \{0,1\}^n$ such that $|\mathcal{F} Δ\mathcal{G}| \leq Cl$, where $C$ is an absolute constant. This is best-possible, up to the value of $C$. Our result can be seen as a `stability' version of the edge isoperimetric inequality in the discrete cube, and as a discrete analogue of the seminal stability result of Fusco, Maggi and Pratelli concerning the isoperimetric inequality in Euclidean space.

math.CO↗

arXiv2018

Stability for the Complete Intersection Theorem, and the Forbidden Intersection Problem of Erdős and Sós

A family $F$ of sets is said to be $t$-intersecting if $|A \cap B| \geq t$ for any $A,B \in F$. The seminal Complete Intersection Theorem of Ahlswede and Khachatrian (1997) gives the maximal size $f(n,k,t)$ of a $t$-intersecting family of $k$-element subsets of $[n]=\{1,2,\ldots,n\}$, together with a characterisation of the extremal families. The forbidden intersection problem, posed by Erdős and Sós in 1971, asks for a determination of the maximal size $g(n,k,t)$ of a family $F$ of $k$-element subsets of $[n]$ such that $|A \cap B| \neq t-1$ for any $A,B \in F$. In this paper, we show that for any fixed $t \in \mathbb{N}$, if $o(n) \leq k \leq n/2-o(n)$, then $g(n,k,t)=f(n,k,t)$. In combination with prior results, this solves the above problem of Erdős and Sós for any constant $t$, except for in the ranges $n/2-o(n) < k < n/2+t/2$ and $k < 2t$. One key ingredient of the proof is the following sharp `stability' result for the Complete Intersection Theorem: if $k/n$ is bounded away from $0$ and $1/2$, and $F$ is a $t$-intersecting family of $k$-element subsets of $[n]$ such that $|F| \geq f(n,k,t) - O(\binom{n-d}{k})$, then there exists a family $G$ such that $G$ is extremal for the Complete Intersection Theorem, and $|F \setminus G| = O(\binom{n-d}{k-d})$. We believe this result to be of interest in its own right; indeed, it proves a conjecture of Friedgut from 2008. We prove it by combining classical `shifting' arguments with a `bootstrapping' method based upon an isoperimetric inequality. Another key ingredient is a `weak regularity lemma' for families of $k$-element subsets of $[n]$, where $k/n$ is bounded away from 0 and 1. This states that any such family $F$ is approximately contained within a `junta', such that the restriction of $F$ to each subcube determined by the junta is `pseudorandom' in a certain sense.

math.CO↗

arXiv2018

On a biased edge isoperimetric inequality for the discrete cube

The `full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary $\partial A$ of a set $A \subset \{0,1\}^n$, as a function of $|A|$. A weaker (but more widely-used) lower bound is $|\partial A| \geq |A| \log_2(2^n/|A|)$, where equality holds iff $A$ is a subcube. In 2011, the first author obtained a sharp `stability' version of the latter result, proving that if $|\partial A| \leq |A| (\log(2^n/|A|)+ε)$, then there exists a subcube $C$ such that $|A ΔC|/|A| = O(ε/\log(1/ε))$. The `weak' version of the edge isoperimetric inequality has the following well-known generalization for the `$p$-biased' measure $μ_p$ on the discrete cube: if $p \leq 1/2$, or if $0 < p < 1$ and $A$ is monotone increasing, then $pμ_p(\partial A) \geq μ_p(A) \log_p(μ_p(A))$. In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if $pμ_p(\partial A) \leq μ_p(A) (\log_p(μ_p(A))+ε)$, then there exists a subcube $C$ such that $μ_p(A ΔC)/μ_p(A) = O(ε' /\log(1/ε'))$, where $ε' =ε\ln (1/p)$. This result is a central component in recent work of the authors proving sharp stability versions of a number of Erdős-Ko-Rado type theorems in extremal combinatorics, including the seminal `complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the `full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the `full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem.

math.CO↗

arXiv2017

Approximation of biased Boolean functions of small total influence by DNF's

The influence of the $k$'th coordinate on a Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ is the probability that flipping $x_k$ changes the value $f(x)$. The total influence $I(f)$ is the sum of influences of the coordinates. The well-known `Junta Theorem' of Friedgut (1998) asserts that if $I(f) \leq M$, then $f$ can be $ε$-approximated by a function that depends on $O(2^{M/ε})$ coordinates. Friedgut's theorem has a wide variety of applications in mathematics and theoretical computer science. For a biased function with $E[f]=μ$, the edge isoperimetric inequality on the cube implies that $I(f) \geq 2μ\log(1/μ)$. Kahn and Kalai (2006) asked, in the spirit of the Junta theorem, whether any $f$ such that $I(f)$ is within a constant factor of the minimum, can be $εμ$-approximated by a DNF of a `small' size (i.e., a union of a small number of sub-cubes). We answer the question by proving the following structure theorem: If $I(f) \leq 2μ(\log(1/μ)+M)$, then $f$ can be $εμ$-approximated by a DNF of size $2^{2^{O(M/ε)}}$. The dependence on $M$ is sharp up to the constant factor in the double exponent.

math.CO↗

arXiv2016

Chvátal's Conjecture and Correlation Inequalities

Chvátal's conjecture in extremal combinatorics asserts that for any decreasing family $\mathcal{F}$ of subsets of a finite set $S$, there is a largest intersecting subfamily of $\mathcal{F}$ consisting of all members of $\mathcal{F}$ that include a particular $x \in S$. In this paper we reformulate the conjecture in terms of influences of variables on Boolean functions and correlation inequalities, and study special cases and variants using tools from discrete Fourier analysis.

math.CO↗

arXiv2015

On the Correlation of Increasing Families

The classical correlation inequality of Harris asserts that any two monotone increasing families on the discrete cube are nonnegatively correlated. In 1996, Talagrand established a lower bound on the correlation in terms of how much the two families depend simultaneously on the same coordinates. Talagrand's method and results inspired a number of important works in combinatorics and probability theory. In this paper we present stronger correlation lower bounds that hold when the increasing families satisfy natural regularity or symmetry conditions. In addition, we present several new classes of examples for which Talagrand's bound is tight. A central tool in the paper is a simple lemma asserting that for monotone events noise decreases correlation. This lemma gives also a very simple derivation of the classical FKG inequality for product measures, and leads to a simplification of part of Talagrand's proof.

math.CO↗

arXiv2015

Juntas in the $\ell^{1}$-grid and Lipschitz maps between discrete tori

We show that if $A \subset [k]^n$, then $A$ is $ε$-close to a junta depending upon at most $\exp(O(|\partial A|/(k^{n-1}ε)))$ coordinates, where $\partial A$ denotes the edge-boundary of $A$ in the $\ell^1$-grid. This is sharp up to the value of the absolute constant in the exponent. This result can be seen as a generalisation of the Junta theorem for the discrete cube, from [E. Friedgut, Boolean functions with low average sensitivity depend on few coordinates, Combinatorica 18 (1998), 27-35], or as a characterization of large subsets of the $\ell^1$-grid whose edge-boundary is small. We use it to prove a result on the structure of Lipschitz functions between two discrete tori; this can be seen as a discrete, quantitative analogue of a recent result of Austin [T. Austin, On the failure of concentration for the $\ell^{\infty}$-ball, preprint]. We also prove a refined version of our junta theorem, which is sharp in a wider range of cases.

math.CO↗

arXiv2015

On the sum of the L1 influences of bounded functions

Let $f\colon \{-1,1\}^n \to [-1,1]$ have degree $d$ as a multilinear polynomial. It is well-known that the total influence of $f$ is at most $d$. Aaronson and Ambainis asked whether the total $L_1$ influence of $f$ can also be bounded as a function of $d$. Bačkurs and Bavarian answered this question in the affirmative, providing a bound of $O(d^3)$ for general functions and $O(d^2)$ for homogeneous functions. We improve on their results by providing a bound of $d^2$ for general functions and $O(d\log d)$ for homogeneous functions. In addition, we prove a bound of $d/(2 π)+o(d)$ for monotone functions, and provide a matching example.

cs.CC↗

arXiv2012

Geometric Influences II: Correlation Inequalities and Noise Sensitivity

In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new definition in Gaussian space. In this paper we prove Gaussian analogues of two of the central applications of influences: Talagrand's lower bound on the correlation of increasing subsets of the discrete cube, and the Benjamini-Kalai-Schramm (BKS) noise sensitivity theorem. We then use the Gaussian results to obtain analogues of Talagrand's bound for all discrete probability spaces and to reestablish analogues of the BKS theorem for biased two-point product spaces.

math.PR↗

arXiv2012

Geometric influences

We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogs of the Kahn-Kalai-Linial (KKL) and Talagrand's influence sum bounds for the new definition. We further prove an analog of a result of Friedgut showing that sets with small "influence sum" are essentially determined by a small number of coordinates. In particular, we establish the following tight analog of the KKL bound: for any set in $\mathbb{R}^n$ of Gaussian measure $t$, there exists a coordinate $i$ such that the $i$th geometric influence of the set is at least $ct(1-t)\sqrt{\log n}/n$, where $c$ is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on $\mathbb{R}^n$ and the class of sets invariant under transitive permutation group of the coordinates.

math.PR↗

arXiv2011

A Quantitative Version of the Gibbard-Satterthwaite Theorem for Three Alternatives

The Gibbard-Satterthwaite theorem states that every non-dictatorial election rule among at least three alternatives can be strategically manipulated. We prove a quantitative version of the Gibbard-Satterthwaite theorem: a random manipulation by a single random voter will succeed with a non-negligible probability for any election rule among three alternatives that is far from being a dictatorship and from having only two alternatives in its range.

math.CO↗

arXiv2011

A Note on the Entropy/Influence Conjecture

The entropy/influence conjecture, raised by Friedgut and Kalai in 1996, seeks to relate two different measures of concentration of the Fourier coefficients of a Boolean function. Roughly saying, it claims that if the Fourier spectrum is "smeared out", then the Fourier coefficients are concentrated on "high" levels. In this note we generalize the conjecture to biased product measures on the discrete cube, and prove a variant of the conjecture for functions with an extremely low Fourier weight on the "high" levels.

math.CO↗

arXiv2010

A simple reduction from a biased measure on the discrete cube to the uniform measure

We show that certain statements related to the Fourier-Walsh expansion of functions with respect to a biased measure on the discrete cube can be deduced from the respective results for the uniform measure by a simple reduction. In particular, we present simple generalizations to the biased measure $μ_p$ of the Bonami-Beckner hypercontractive inequality, and of Talagrand's lower bound on the size of the boundary of subsets of the discrete cube. Our generalizations are tight up to constant factors.

math.CO↗

arXiv2010

A tight quantitative version of Arrow's impossibility theorem

The well-known Impossibility Theorem of Arrow asserts that any Generalized Social Welfare Function (GSWF) with at least three alternatives, which satisfies Independence of Irrelevant Alternatives (IIA) and Unanimity and is not a dictatorship, is necessarily non-transitive. In 2002, Kalai asked whether one can obtain the following quantitative version of the theorem: For any $ε>0$, there exists $δ=δ(ε)$ such that if a GSWF on three alternatives satisfies the IIA condition and its probability of non-transitive outcome is at most $δ$, then the GSWF is at most $ε$-far from being a dictatorship or from breaching the Unanimity condition. In 2009, Mossel proved such quantitative version, with $δ(ε)=\exp(-C/ε^{21})$, and generalized it to GSWFs with $k$ alternatives, for all $k \geq 3$. In this paper we show that the quantitative version holds with $δ(ε)=C \cdot ε^3$, and that this result is tight up to logarithmic factors. Furthermore, our result (like Mossel's) generalizes to GSWFs with $k$ alternatives. Our proof is based on the works of Kalai and Mossel, but uses also an additional ingredient: a combination of the Bonami-Beckner hypercontractive inequality with a reverse hypercontractive inequality due to Borell, applied to find simultaneously upper bounds and lower bounds on the "noise correlation" between Boolean functions on the discrete cube.

math.CO↗

arXiv2010

Quantitative relation between noise sensitivity and influences

A Boolean function $f:\{0,1\}^n \to \{0,1\}$ is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm showed that if the sum of squares of influences in $f$ is close to zero then $f$ must be noise sensitive. We show a quantitative version of this result which does not depend on $n$, and prove that it is tight for certain parameters. Our results hold also for a general product measure $μ_p$ on the discrete cube, as long as $\log 1/p \ll \log n$. We note that in [BKS], a quantitative relation between the sum of squares of the influences and the noise sensitivity was also shown, but only when the sum of squares is bounded by $n^{-c}$ for a constant $c$. Our results require a generalization of a lemma of Talagrand on the Fourier coefficients of monotone Boolean functions. In order to achieve it, we present a considerably shorter proof of Talagrand's lemma, which easily generalizes in various directions, including non-monotone functions.

math.CO↗

arXiv2009

On The Probability of a Rational Outcome for Generalized Social Welfare Functions on Three Alternatives

In [G. Kalai, A Fourier-theoretic Perspective on the Condorcet Paradox and Arrow's Theorem, Adv. in Appl. Math. 29(3) (2002), pp. 412--426], Kalai investigated the probability of a rational outcome for a generalized social welfare function (GSWF) on three alternatives, when the individual preferences are uniform and independent. In this paper we generalize Kalai's results to a broader class of distributions of the individual preferences, and obtain new lower bounds on the probability of a rational outcome in several classes of GSWFs. In particular, we show that if the GSWF is monotone and balanced and the distribution of the preferences is uniform, then the probability of a rational outcome is at least 3/4, proving a conjecture raised by Kalai. The tools used in the paper are analytic: the Fourier-Walsh expansion of Boolean functions on the discrete cube, properties of the Bonamie-Beckner noise operator, and the FKG inequality.

math.CO↗

arXiv2009

On The Influences of Variables on Boolean Functions in Product Spaces

In this paper we consider the influences of variables on Boolean functions in general product spaces. Unlike the case of functions on the discrete cube where there is a clear definition of influence, in the general case at least three definitions were presented in different papers. We propose a family of definitions for the influence, that contains all the known definitions, as well as other natural definitions, as special cases. We prove a generalization of the BKKKL theorem, which is tight in terms of the definition of influence used in the assertion, and use it to generalize several known results on influences in general product spaces.

math.CO↗

arXiv2006

MV3: A new word based stream cipher using rapid mixing and revolving buffers

MV3 is a new word based stream cipher for encrypting long streams of data. A direct adaptation of a byte based cipher such as RC4 into a 32- or 64-bit word version will obviously need vast amounts of memory. This scaling issue necessitates a look for new components and principles, as well as mathematical analysis to justify their use. Our approach, like RC4's, is based on rapidly mixing random walks on directed graphs (that is, walks which reach a random state quickly, from any starting point). We begin with some well understood walks, and then introduce nonlinearity in their steps in order to improve security and show long term statistical correlations are negligible. To minimize the short term correlations, as well as to deter attacks using equations involving successive outputs, we provide a method for sequencing the outputs derived from the walk using three revolving buffers. The cipher is fast -- it runs at a speed of less than 5 cycles per byte on a Pentium IV processor. A word based cipher needs to output more bits per step, which exposes more correlations for attacks. Moreover we seek simplicity of construction and transparent analysis. To meet these requirements, we use a larger state and claim security corresponding to only a fraction of it. Our design is for an adequately secure word-based cipher; our very preliminary estimate puts the security close to exhaustive search for keys of size < 256 bits.

cs.CR↗