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Oded Regev

Publications and source records attributed to Oded Regev.

At least 37 records · Page 2Linked to original sources
arXiv2018

Kneser graphs are like Swiss cheese

We prove that for a large family of product graphs, and for Kneser graphs $K(n,αn)$ with fixed $α<1/2$, the following holds. Any set of vertices that spans a small proportion of the edges in the graph can be made independent by removing a small proportion of the vertices of the graph. This allows us to strengthen the results of [DinurFR06] and [DinurF09], and show that any independent set in these graphs is almost contained in an independent set which depends on few coordinates. Our proof is inspired by, and follows some of the main ideas of, Fox's proof of the graph removal lemma [Fox11].

math.CO↗
arXiv2017

On Learning Mixtures of Well-Separated Gaussians

We consider the problem of efficiently learning mixtures of a large number of spherical Gaussians, when the components of the mixture are well separated. In the most basic form of this problem, we are given samples from a uniform mixture of $k$ standard spherical Gaussians, and the goal is to estimate the means up to accuracy $δ$ using $poly(k,d, 1/δ)$ samples. In this work, we study the following question: what is the minimum separation needed between the means for solving this task? The best known algorithm due to Vempala and Wang [JCSS 2004] requires a separation of roughly $\min\{k,d\}^{1/4}$. On the other hand, Moitra and Valiant [FOCS 2010] showed that with separation $o(1)$, exponentially many samples are required. We address the significant gap between these two bounds, by showing the following results. 1. We show that with separation $o(\sqrt{\log k})$, super-polynomially many samples are required. In fact, this holds even when the $k$ means of the Gaussians are picked at random in $d=O(\log k)$ dimensions. 2. We show that with separation $Ω(\sqrt{\log k})$, $poly(k,d,1/δ)$ samples suffice. Note that the bound on the separation is independent of $δ$. This result is based on a new and efficient "accuracy boosting" algorithm that takes as input coarse estimates of the true means and in time $poly(k,d, 1/δ)$ outputs estimates of the means up to arbitrary accuracy $δ$ assuming the separation between the means is $Ω(\min\{\sqrt{\log k},\sqrt{d}\})$ (independently of $δ$). We also present a computationally efficient algorithm in $d=O(1)$ dimensions with only $Ω(\sqrt{d})$ separation. These results together essentially characterize the optimal order of separation between components that is needed to learn a mixture of $k$ spherical Gaussians with polynomial samples.

cs.DS↗
arXiv2017

A Sharp Tail Bound for the Expander Random Sampler

Consider an expander graph in which a $μ$ fraction of the vertices are marked. A random walk starts at a uniform vertex and at each step continues to a random neighbor. Gillman showed in 1993 that the number of marked vertices seen in a random walk of length $n$ is concentrated around its expectation, $Φ:= μn$, independent of the size of the graph. Here we provide a new and sharp tail bound, improving on the existing bounds whenever $μ$ is not too large.

math.PR↗
arXiv2017

A counterexample to a strong variant of the Polynomial Freiman-Ruzsa conjecture in Euclidean space

The Polynomial Freiman-Ruzsa conjecture is one of the central open problems in additive combinatorics. If true, it would give tight quantitative bounds relating combinatorial and algebraic notions of approximate subgroups. In this note, we restrict our attention to subsets of Euclidean space. In this regime, the original conjecture considers approximate algebraic subgroups as the set of lattice points in a convex body. Green asked in 2007 whether this can be simplified to a generalized arithmetic progression, while not losing more than a polynomial factor in the underlying parameters. We give a negative answer to this question, based on a recent reverse Minkowski theorem combined with estimates for random lattices.

math.NT↗
arXiv2017

The Minrank of Random Graphs

The minrank of a graph $G$ is the minimum rank of a matrix $M$ that can be obtained from the adjacency matrix of $G$ by switching some ones to zeros (i.e., deleting edges) and then setting all diagonal entries to one. This quantity is closely related to the fundamental information-theoretic problems of (linear) index coding (Bar-Yossef et al., FOCS'06), network coding and distributed storage, and to Valiant's approach for proving superlinear circuit lower bounds (Valiant, Boolean Function Complexity '92). We prove tight bounds on the minrank of random Erdős-Rényi graphs $G(n,p)$ for all regimes of $p\in[0,1]$. In particular, for any constant $p$, we show that $\mathsf{minrk}(G) = Θ(n/\log n)$ with high probability, where $G$ is chosen from $G(n,p)$. This bound gives a near quadratic improvement over the previous best lower bound of $Ω(\sqrt{n})$ (Haviv and Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky and Stav (FOCS '07). Our lower bound matches the well-known upper bound obtained by the "clique covering" solution, and settles the linear index coding problem for random graphs. Finally, our result suggests a new avenue of attack, via derandomization, on Valiant's approach for proving superlinear lower bounds for logarithmic-depth semilinear circuits.

cs.CC↗
arXiv2016

An Inequality for Gaussians on Lattices

$ \newcommand{\R}{\ensuremath{\mathbb{R}}} \newcommand{\lat}{\mathcal{L}} \newcommand{\ensuremath}[1]{#1} $We show that for any lattice $\lat \subseteq \R^n$ and vectors $\vec{x}, \vec{y} \in \R^n$, \[ ρ(\lat + \vec{x})^2 ρ(\lat + \vec{y})^2 \leq ρ(\lat)^2 ρ(\lat + \vec{x} + \vec{y}) ρ(\lat + \vec{x} - \vec{y}) \; , \] where $ρ$ is the Gaussian measure $ρ(A) := \sum_{\vec{w} \in A} \exp(-π\| \vec{w} \|^2)$. We show a number of applications, including bounds on the moments of the discrete Gaussian distribution, various monotonicity properties of the heat kernel on flat tori, and a positive correlation inequality for Gaussian measures on lattices.

math.PR↗
arXiv2016

A Note on Koldobsky's Lattice Slicing Inequality

$ \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \renewcommand{\vec}[1]{\mathbf{#1}} $We show that if $K \subset \R^d$ is an origin-symmetric convex body, then there exists a vector $\vec{y} \in \Z^d$ such that \begin{align*} |K \cap \Z^d \cap \vec{y}^\perp| / |K \cap \Z^d| \ge \min(1,c \cdot d^{-1} \cdot \mathrm{vol}(K)^{-1/(d-1)}) \; , \end{align*} for some absolute constant $c> 0$, where $\vec{y}^\perp$ denotes the subspace orthogonal to $\vec{y}$. This gives a partial answer to a question by Koldobsky.

math.MG↗
arXiv2016

Towards Strong Reverse Minkowski-type Inequalities for Lattices

We present a natural reverse Minkowski-type inequality for lattices, which gives upper bounds on the number of lattice points in a Euclidean ball in terms of sublattice determinants, and conjecture its optimal form. The conjecture exhibits a surprising wealth of connections to various areas in mathematics and computer science, including a conjecture motivated by integer programming by Kannan and Lovász (Annals of Math. 1988), a question from additive combinatorics asked by Green, a question on Brownian motions asked by Saloff-Coste (Colloq. Math. 2010), a theorem by Milman and Pisier from convex geometry (Ann. Probab. 1987), worst-case to average-case reductions in lattice-based cryptography, and more. We present these connections, provide evidence for the conjecture, and discuss possible approaches towards a proof. Our main technical contribution is in proving that our conjecture implies the $\ell_2$ case of the Kannan and Lovász conjecture. The proof relies on a novel convex relaxation for the covering radius, and a rounding procedure for based on "uncrossing" lattice subspaces.

math.MG↗
arXiv2015

A Counterexample to Monotonicity of Relative Mass in Random Walks

For a finite undirected graph $G = (V,E)$, let $p_{u,v}(t)$ denote the probability that a continuous-time random walk starting at vertex $u$ is in $v$ at time $t$. In this note we give an example of a Cayley graph $G$ and two vertices $u,v \in G$ for which the function \[ r_{u,v}(t) = \frac{p_{u,v}(t)}{p_{u,u}(t)} \qquad t \geq 0 \] is not monotonically non-decreasing. This answers a question asked by Peres in 2013.

math.PR↗
arXiv2015

The Restricted Isometry Property of Subsampled Fourier Matrices

A matrix $A \in \mathbb{C}^{q \times N}$ satisfies the restricted isometry property of order $k$ with constant $\varepsilon$ if it preserves the $\ell_2$ norm of all $k$-sparse vectors up to a factor of $1\pm \varepsilon$. We prove that a matrix $A$ obtained by randomly sampling $q = O(k \cdot \log^2 k \cdot \log N)$ rows from an $N \times N$ Fourier matrix satisfies the restricted isometry property of order $k$ with a fixed $\varepsilon$ with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).

cs.DS↗
arXiv2015

Solving the Shortest Vector Problem in $2^n$ Time via Discrete Gaussian Sampling

We give a randomized $2^{n+o(n)}$-time and space algorithm for solving the Shortest Vector Problem (SVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm: the deterministic $\widetilde{O}(4^n)$-time and $\widetilde{O}(2^n)$-space algorithm of Micciancio and Voulgaris (STOC 2010, SIAM J. Comp. 2013). In fact, we give a conceptually simple algorithm that solves the (in our opinion, even more interesting) problem of discrete Gaussian sampling (DGS). More specifically, we show how to sample $2^{n/2}$ vectors from the discrete Gaussian distribution at any parameter in $2^{n+o(n)}$ time and space. (Prior work only solved DGS for very large parameters.) Our SVP result then follows from a natural reduction from SVP to DGS. We also show that our DGS algorithm implies a $2^{n + o(n)}$-time algorithm that approximates the Closest Vector Problem to within a factor of $1.97$. In addition, we give a more refined algorithm for DGS above the so-called smoothing parameter of the lattice, which can generate $2^{n/2}$ discrete Gaussian samples in just $2^{n/2+o(n)}$ time and space. Among other things, this implies a $2^{n/2+o(n)}$-time and space algorithm for $1.93$-approximate decision SVP.

cs.DS↗
arXiv2015

Beating the random assignment on constraint satisfaction problems of bounded degree

We show that for any odd $k$ and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a $\frac{1}{2} + Ω(1/\sqrt{D})$ fraction of constraints, where $D$ is a bound on the number of constraints that each variable occurs in. This improves both qualitatively and quantitatively on the recent work of Farhi, Goldstone, and Gutmann (2014), which gave a \emph{quantum} algorithm to find an assignment satisfying a $\frac{1}{2} + Ω(D^{-3/4})$ fraction of the equations. For arbitrary constraint satisfaction problems, we give a similar result for "triangle-free" instances; i.e., an efficient algorithm that finds an assignment satisfying at least a $μ+ Ω(1/\sqrt{D})$ fraction of constraints, where $μ$ is the fraction that would be satisfied by a uniformly random assignment.

cs.CC↗
arXiv2015

Efficient Quantum Algorithms for (Gapped) Group Testing and Junta Testing

In the $k$-junta testing problem, a tester has to efficiently decide whether a given function $f:\{0,1\}^n\rightarrow \{0,1\}$ is a $k$-junta (i.e., depends on at most $k$ of its input bits) or is $ε$-far from any $k$-junta. Our main result is a quantum algorithm for this problem with query complexity $\tilde O(\sqrt{k/ε})$ and time complexity $\tilde O(n\sqrt{k/ε})$. This quadratically improves over the query complexity of the previous best quantum junta tester, due to Atıcı and Servedio. Our tester is based on a new quantum algorithm for a gapped version of the combinatorial group testing problem, with an up to quartic improvement over the query complexity of the best classical algorithm. For our upper bound on the time complexity we give a near-linear time implementation of a shallow variant of the quantum Fourier transform over the symmetric group, similar to the Schur-Weyl transform. We also prove a lower bound of $Ω(k^{1/3})$ queries for junta-testing (for constant $ε$).

cs.CC↗
arXiv2015

The List-Decoding Size of Fourier-Sparse Boolean Functions

A function defined on the Boolean hypercube is $k$-Fourier-sparse if it has at most $k$ nonzero Fourier coefficients. For a function $f: \mathbb{F}_2^n \rightarrow \mathbb{R}$ and parameters $k$ and $d$, we prove a strong upper bound on the number of $k$-Fourier-sparse Boolean functions that disagree with $f$ on at most $d$ inputs. Our bound implies that the number of uniform and independent random samples needed for learning the class of $k$-Fourier-sparse Boolean functions on $n$ variables exactly is at most $O(n \cdot k \log k)$. As an application, we prove an upper bound on the query complexity of testing Booleanity of Fourier-sparse functions. Our bound is tight up to a logarithmic factor and quadratically improves on a result due to Gur and Tamuz (Chicago J. Theor. Comput. Sci., 2013).

cs.DS↗
arXiv2014

On the Closest Vector Problem with a Distance Guarantee

We present a substantially more efficient variant, both in terms of running time and size of preprocessing advice, of the algorithm by Liu, Lyubashevsky, and Micciancio for solving CVPP (the preprocessing version of the Closest Vector Problem, CVP) with a distance guarantee. For instance, for any $α< 1/2$, our algorithm finds the (unique) closest lattice point for any target point whose distance from the lattice is at most $α$ times the length of the shortest nonzero lattice vector, requires as preprocessing advice only $N \approx \widetilde{O}(n \exp(α^2 n /(1-2α)^2))$ vectors, and runs in time $\widetilde{O}(nN)$. As our second main contribution, we present reductions showing that it suffices to solve CVP, both in its plain and preprocessing versions, when the input target point is within some bounded distance of the lattice. The reductions are based on ideas due to Kannan and a recent sparsification technique due to Dadush and Kun. Combining our reductions with the LLM algorithm gives an approximation factor of $O(n/\sqrt{\log n})$ for search CVPP, improving on the previous best of $O(n^{1.5})$ due to Lagarias, Lenstra, and Schnorr. When combined with our improved algorithm we obtain, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve CVPP with (the only slightly worse) approximation factor of O(n).

cs.DS↗
arXiv2014

A Note on Discrete Gaussian Combinations of Lattice Vectors

We analyze the distribution of $\sum_{i=1}^m v_i \bx_i$ where $\bx_1,...,\bx_m$ are fixed vectors from some lattice $\cL \subset \R^n$ (say $\Z^n$) and $v_1,...,v_m$ are chosen independently from a discrete Gaussian distribution over $\Z$. We show that under a natural constraint on $\bx_1,...,\bx_m$, if the $v_i$ are chosen from a wide enough Gaussian, the sum is statistically close to a discrete Gaussian over $\cL$. We also analyze the case of $\bx_1,...,\bx_m$ that are themselves chosen from a discrete Gaussian distribution (and fixed). Our results simplify and qualitatively improve upon a recent result by Agrawal, Gentry, Halevi, and Sahai \cite{AGHS13}.

cs.CR↗
arXiv2013

On the Lattice Isomorphism Problem

We study the Lattice Isomorphism Problem (LIP), in which given two lattices L_1 and L_2 the goal is to decide whether there exists an orthogonal linear transformation mapping L_1 to L_2. Our main result is an algorithm for this problem running in time n^{O(n)} times a polynomial in the input size, where n is the rank of the input lattices. A crucial component is a new generalized isolation lemma, which can isolate n linearly independent vectors in a given subset of Z^n and might be useful elsewhere. We also prove that LIP lies in the complexity class SZK.

cs.DS↗