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The Gaussian Surface Area and Noise Sensitivity of Degree-$d$ PolynomialsDec 14 2009We provide asymptotically sharp bounds for the Gaussian surface area and the Gaussian noise sensitivity of polynomial threshold functions. In particular we show that if $f$ is a degree-$d$ polynomial threshold function, then its Gaussian sensitivity at ... More

An Asymptotic for the Number of Solutions to Linear Equations in Prime Numbers from Specified Chebotarev ClassesSep 07 2010Nov 05 2012We extend known results on the number of solutions to a linear equation in at least three prime numbers when the primes involved are required to lie in specified Chebotarev classes. We prove asymptotic results similar to previous ones only now taking ... More

A Structure Theorem for Poorly Anticoncentrated Gaussian Chaoses and Applications to the Study of Polynomial Threshold FunctionsApr 02 2012Aug 15 2012We prove a structural result for degree-$d$ polynomials. In particular, we show that any degree-$d$ polynomial, $p$ can be approximated by another polynomial, $p_0$, which can be decomposed as some function of polynomials $q_1,...,q_m$ with $q_i$ normalized ... More

A Small PRG for Polynomial Threshold Functions of GaussiansApr 06 2011We develop a pseudo-random generator to fool degree-$d$ polynomial threshold functions with respect to the Gaussian distribution. For $c>0$ any constant, we construct a pseudo-random generator that fools such functions to within $\epsilon$ and has seed ... More

The Correct Exponent for the Gotsman-Linial ConjectureOct 04 2012We prove a new bound on the average sensitivity of polynomial threshold functions. In particular we show that a polynomial threshold function of degree $d$ in at most $n$ variables has average sensitivity at most $\sqrt{n}(\log(n))^{O(d\log(d))}2^{O(d^2\log(d)}$. ... More

Canonical Projective Embeddings of the Deligne-Lusztig Curves Associated to $2A2$, $2B2$ and $2G2$Dec 30 2010The Deligne-Lusztig varieties associated to the Coxeter classes of the algebraic groups 2A2, 2B2 and 2G2 are affine algebraic curves. We produce explicit projective models of the closures of these curves. Furthermore for $d$ the Coxeter number of these ... More

A Low-Depth Monotone Function that is not an Approximate JuntaJun 27 2012Jun 14 2013We provide an example of a monotone Boolean function on the hypercube given by a low depth decision tree that is not well approximated by any k-junta for small k.

On the Crossing Number of Complete Graphs with an Uncrossed Hamiltonian CycleSep 11 2013We prove new lower bounds on the crossing number of a complete graphs assuming that it is drawn in such a way that it contains a Hamiltonian cycle with no crossings.

The Average Sensitivity of an Intersection of Half SpacesSep 11 2013Jan 27 2015We prove new bounds on the average sensitivity of the indicator function of an intersection of $k$ halfspaces. In particular, we prove the optimal bound of $O(\sqrt{n\log(k)})$. This generalizes a result of Nazarov, who proved the analogous result in ... More

On the Number of ABC Solutions with Restricted Radical SizesApr 13 2011Sep 15 2014We consider a variant of the ABC Conjecture, attempting to count the number of solutions to $A+B+C=0$, in relatively prime integers $A,B,C$ each of absolute value less than $N$ with $r(A)<|A|^a, r(B)<|B|^b, r(C)<|C|^c.$ The ABC Conjecture is equivalent ... More

$k$-Independent Gaussians Fool Polynomial Threshold FunctionsDec 07 2010Nov 11 2011We show that any $O_d(\epsilon^{-4d 7^d})$-independent family of Gaussians $\epsilon$-fools any degree-$d$ polynomial threshold function.

On the Ranks of the 2-Selmer Groups of Twists of a Given Elliptic CurveSep 07 2010Sep 03 2013We extend work of Swinnerton-Dyer on the density of the number of twists of a given elliptic curve that have 2-Selmer group of a particular rank.

Unary Subset-Sum is in LogspaceDec 06 2010Dec 08 2010We present a simple Logspace algorithm that solves the Unary Subset-Sum problem.

Small Designs for Path Connected Spaces and Path Connected Homogeneous SpacesDec 21 2011Jun 25 2012We prove the existence of designs of small size in a number of contexts. In particular our techniques can be applied to prove the existence of $n$-designs on $S^{d}$ of size $O_d(n^{d}\log(n)^{d-1})$.

A Pseudorandom Generator for Polynomial Threshold Functions of Gaussian with Subpolynomial Seed LengthOct 04 2012We develop a pseudorandom generator that fools degree-$d$ polynomial threshold functions in $n$ variables with respect to the Gaussian distribution and has seed length $O_{c,d}(\log(n) \epsilon^{-c})$.

A Polylogarithmic PRG for Degree $2$ Threshold Functions in the Gaussian SettingApr 03 2014We devise a new pseudorandom generator against degree 2 polynomial threshold functions in the Gaussian setting. We manage to achieve $\epsilon$ error with seed length polylogarithmic in $\epsilon$ and the dimension, and exponential improvement over previous ... More

Quantum Money from Modular FormsSep 16 2018Oct 16 2018We present a new idea for a class of public key quantum money protocols where the bills are joint eigenstates of systems of commuting unitary operators. We show that this system is secure against black box attacks, and propose an implementation where ... More

Sparser Johnson-Lindenstrauss TransformsDec 07 2010Feb 05 2014We give two different and simple constructions for dimensionality reduction in $\ell_2$ via linear mappings that are sparse: only an $O(\varepsilon)$-fraction of entries in each column of our embedding matrices are non-zero to achieve distortion $1+\varepsilon$ ... More

A Derandomized Sparse Johnson-Lindenstrauss TransformJun 18 2010Dec 07 2010Recent work of [Dasgupta-Kumar-Sarlos, STOC 2010] gave a sparse Johnson-Lindenstrauss transform and left as a main open question whether their construction could be efficiently derandomized. We answer their question affirmatively by giving an alternative ... More

Degree-$d$ Chow Parameters Robustly Determine Degree-$d$ PTFs (and Algorithmic Applications)Nov 07 2018The degree-$d$ Chow parameters of a Boolean function $f: \{-1,1\}^n \to \mathbb{R}$ are its degree at most $d$ Fourier coefficients. It is well-known that degree-$d$ Chow parameters uniquely characterize degree-$d$ polynomial threshold functions (PTFs) ... More

Super-Linear Gate and Super-Quadratic Wire Lower Bounds for Depth-Two and Depth-Three Threshold CircuitsNov 24 2015In order to formally understand the power of neural computing, we first need to crack the frontier of threshold circuits with two and three layers, a regime that has been surprisingly intractable to analyze. We prove the first super-linear gate lower ... More

A New Approach for Testing Properties of Discrete DistributionsJan 21 2016May 09 2016In this work, we give a novel general approach for distribution testing. We describe two techniques: our first technique gives sample-optimal testers, while our second technique gives matching sample lower bounds. As a consequence, we resolve the sample ... More

Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian MixturesNov 10 2016We prove the first {\em Statistical Query lower bounds} for two fundamental high-dimensional learning problems involving Gaussian distributions: (1) learning Gaussian mixture models (GMMs), and (2) robust (agnostic) learning of a single unknown mean Gaussian. ... More

Sharp Bounds for Generalized Uniformity TestingSep 07 2017We study the problem of generalized uniformity testing \cite{BC17} of a discrete probability distribution: Given samples from a probability distribution $p$ over an {\em unknown} discrete domain $\mathbf{\Omega}$, we want to distinguish, with probability ... More

Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian MixturesNov 10 2016May 17 2017We describe a general technique that yields the first {\em Statistical Query lower bounds} for a range of fundamental high-dimensional learning problems involving Gaussian distributions. Our main results are for the problems of (1) learning Gaussian mixture ... More

Bounded Independence Fools Degree-2 Threshold FunctionsNov 17 2009Feb 18 2010Let x be a random vector coming from any k-wise independent distribution over {-1,1}^n. For an n-variate degree-2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive epsilon for k = poly(1/epsilon). This answers an open question of ... More

Testing Identity of Multidimensional HistogramsApr 10 2018Feb 19 2019We investigate the problem of identity testing for multidimensional histogram distributions. A distribution $p: D \rightarrow \mathbb{R}_+$, where $D \subseteq \mathbb{R}^d$, is called a $k$-histogram if there exists a partition of the domain into $k$ ... More

Classifying toric and semitoric fans by lifting equations from $\textrm{SL}_2 (\mathbb{Z})$Feb 26 2015Sep 19 2015We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group $\textrm{SL}_2 (\mathbb{Z})$ to its preimage in the universal cover of $\textrm{SL}_2 (\mathbb{R})$. With this method we ... More

Minimal models of compact symplectic semitoric manifoldsOct 18 2016A symplectic semitoric manifold is a symplectic $4$-manifold endowed with a Hamiltonian $(S^1 \times \mathbb{R})$-action satisfying certain conditions. The goal of this paper is to construct a new symplectic invariant of semitoric manifolds, the helix, ... More

Learning Multivariate Log-concave DistributionsMay 26 2016We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on $\mathbb{R}^d$, for all $d \geq 1$. Prior to our work, no upper bound on ... More

Efficient Robust Proper Learning of Log-concave DistributionsJun 09 2016We study the {\em robust proper learning} of univariate log-concave distributions (over continuous and discrete domains). Given a set of samples drawn from an unknown target distribution, we want to compute a log-concave hypothesis distribution that is ... More

List-Decodable Robust Mean Estimation and Learning Mixtures of Spherical GaussiansNov 20 2017We study the problem of list-decodable Gaussian mean estimation and the related problem of learning mixtures of separated spherical Gaussians. We develop a set of techniques that yield new efficient algorithms with significantly improved guarantees for ... More

Minimal models of compact symplectic semitoric manifoldsOct 18 2016Nov 16 2016A symplectic semitoric manifold is a symplectic $4$-manifold endowed with a Hamiltonian $(S^1 \times \mathbb{R})$-action satisfying certain conditions. The goal of this paper is to construct a new symplectic invariant of symplectic semitoric manifolds, ... More

Classifying toric and semitoric fans by lifting equations from $\textrm{SL}_2 (\mathbb{Z})$Feb 26 2015Oct 25 2016We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group $\textrm{SL}_2 (\mathbb{Z})$ to its preimage in the universal cover of $\textrm{SL}_2 (\mathbb{R})$. With this method we ... More

Learning Geometric Concepts with Nasty NoiseJul 05 2017We study the efficient learnability of geometric concept classes - specifically, low-degree polynomial threshold functions (PTFs) and intersections of halfspaces - when a fraction of the data is adversarially corrupted. We give the first polynomial-time ... More

Testing Identity of Structured DistributionsOct 08 2014We study the question of identity testing for structured distributions. More precisely, given samples from a {\em structured} distribution $q$ over $[n]$ and an explicit distribution $p$ over $[n]$, we wish to distinguish whether $q=p$ versus $q$ is at ... More

Asymptotic Improvements of Lower Bounds for the Least Common Multiples of Arithmetic ProgressionsDec 30 2010Jul 01 2014For relatively prime positive integers $u_0$ and $r$, we consider the least common multiple $L_n:=\mathrm{lcm}(u_0,u_1,\ldots, u_n)$ of the finite arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower bounds on $L_n$ which improve upon ... More

The Power of Comparisons for Actively Learning Linear ClassifiersJul 08 2019In the world of big data, large but costly to label datasets dominate many fields. Active learning, an unsupervised alternative to the standard PAC-learning model, was introduced to explore whether adaptive labeling could learn concepts with exponentially ... More

Quantum interpolation of polynomialsSep 30 2009Mar 18 2010We consider quantum interpolation of polynomials. We imagine a quantum computer with black-box access to input/output pairs (x_i, f(x_i)), where f is a degree-d polynomial, and we wish to compute f(0). We give asymptotically tight quantum lower bounds ... More

Near-Optimal Closeness Testing of Discrete Histogram DistributionsMar 06 2017We investigate the problem of testing the equivalence between two discrete histograms. A {\em $k$-histogram} over $[n]$ is a probability distribution that is piecewise constant over some set of $k$ intervals over $[n]$. Histograms have been extensively ... More

Learning Multivariate Log-concave DistributionsMay 26 2016Jun 05 2017We study the problem of estimating multivariate log-concave probability density functions. We prove the first sample complexity upper bound for learning log-concave densities on $\mathbb{R}^d$, for all $d \geq 1$. Prior to our work, no upper bound on ... More

Properly Learning Poisson Binomial Distributions in Almost Polynomial TimeNov 12 2015We give an algorithm for properly learning Poisson binomial distributions. A Poisson binomial distribution (PBD) of order $n$ is the discrete probability distribution of the sum of $n$ mutually independent Bernoulli random variables. Given $\widetilde{O}(1/\epsilon^2)$ ... More

The Fourier Transform of Poisson Multinomial Distributions and its Algorithmic ApplicationsNov 11 2015Jun 22 2016An $(n, k)$-Poisson Multinomial Distribution (PMD) is a random variable of the form $X = \sum_{i=1}^n X_i$, where the $X_i$'s are independent random vectors supported on the set of standard basis vectors in $\mathbb{R}^k.$ In this paper, we obtain a refined ... More

Generalized comparison trees for point-location problemsApr 23 2018Let $H$ be an arbitrary family of hyper-planes in $d$-dimensions. We show that the point-location problem for $H$ can be solved by a linear decision tree that only uses a special type of queries called \emph{generalized comparison queries}. These queries ... More

Waring's Theorem for Binary PowersJan 13 2018A natural number is a binary $k$'th power if its binary representation consists of $k$ consecutive identical blocks. We prove an analogue of Waring's theorem for sums of binary $k$'th powers. More precisely, we show that for each integer $k \geq 2$, there ... More

Optimal Algorithms and Lower Bounds for Testing Closeness of Structured DistributionsAug 22 2015We give a general unified method that can be used for $L_1$ {\em closeness testing} of a wide range of univariate structured distribution families. More specifically, we design a sample optimal and computationally efficient algorithm for testing the equivalence ... More

Optimal Learning via the Fourier Transform for Sums of Independent Integer Random VariablesMay 04 2015Nov 23 2015We study the structure and learnability of sums of independent integer random variables (SIIRVs). For $k \in \mathbb{Z}_{+}$, a $k$-SIIRV of order $n \in \mathbb{Z}_{+}$ is the probability distribution of the sum of $n$ independent random variables each ... More

Near-optimal linear decision trees for k-SUM and related problemsMay 04 2017We construct near optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry. For example, for any constant $k$, we construct linear decision trees that solve the $k$-SUM problem on $n$ elements using $O(n ... More

A proof of Andrews' conjecture on Partitions with no short sequencesApr 20 2012Holroyd, Liggett, and Romik introduced the following probability model. Let $C_1, C_2,...$ be independent events with probabilities $\P_s(C_n)= 1-e^{-ns}$ under a probability measure $\P_s$ with $0<s<1$. Let $A_k$ be the event that there is no sequence ... More

Classifying Toric and Semitoric Fans by Lifting Equations from ${\rm SL}_2({\mathbb Z})$Feb 26 2015Feb 22 2018We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group ${\rm SL}_2({\mathbb Z})$ to its preimage in the universal cover of ${\rm SL}_2({\mathbb R})$. With this method we recover ... More

Nearly Tight Bounds for Robust Proper Learning of Halfspaces with a MarginAug 29 2019We study the problem of {\em properly} learning large margin halfspaces in the agnostic PAC model. In more detail, we study the complexity of properly learning $d$-dimensional halfspaces on the unit ball within misclassification error $\alpha \cdot \mathrm{OPT}_{\gamma} ... More

Active classification with comparison queriesApr 11 2017Jun 02 2017We study an extension of active learning in which the learning algorithm may ask the annotator to compare the distances of two examples from the boundary of their label-class. For example, in a recommendation system application (say for restaurants), ... More

Testing Conditional Independence of Discrete DistributionsNov 30 2017Jul 01 2018We study the problem of testing \emph{conditional independence} for discrete distributions. Specifically, given samples from a discrete random variable $(X, Y, Z)$ on domain $[\ell_1]\times[\ell_2] \times [n]$, we want to distinguish, with probability ... More

Minimal S-universality criteria may vary in sizeJan 29 2011In this note, we give simple examples of sets S of quadratic forms that have minimal S-universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh in the negative.

Communication and Memory Efficient Testing of Discrete DistributionsJun 11 2019We study distribution testing with communication and memory constraints in the following computational models: (1) The {\em one-pass streaming model} where the goal is to minimize the sample complexity of the protocol subject to a memory constraint, and ... More

Learning Ising Models with Independent FailuresFeb 13 2019We give the first efficient algorithm for learning the structure of an Ising model that tolerates independent failures; that is, each entry of the observed sample is missing with some unknown probability p. Our algorithm matches the essentially optimal ... More

Fast Moment Estimation in Data Streams in Optimal SpaceJul 23 2010We give a space-optimal algorithm with update time O(log^2(1/eps)loglog(1/eps)) for (1+eps)-approximating the pth frequency moment, 0 < p < 2, of a length-n vector updated in a data stream. This provides a nearly exponential improvement in the update ... More

Central Limit Theorems for some Set Partition StatisticsFeb 03 2015We prove the conjectured limiting normality for the number of crossings of a uniformly chosen set partition of [n] = {1,2,...,n}. The arguments use a novel stochastic representation and are also used to prove central limit theorems for the dimension index ... More

Revisiting Norm Estimation in Data StreamsNov 21 2008Apr 09 2009The problem of estimating the pth moment F_p (p nonnegative and real) in data streams is as follows. There is a vector x which starts at 0, and many updates of the form x_i <-- x_i + v come sequentially in a stream. The algorithm also receives an error ... More

Closed expressions for averages of set partition statisticsApr 16 2013Jun 10 2013In studying the enumerative theory of super characters' of the group of upper triangular matrices over a finite field we found that the moments (mean, variance and higher moments) of novel statistics on set partitions have simple closed expressions as ... More

Fourier-sparse interpolation without a frequency gapSep 06 2016We consider the problem of estimating a Fourier-sparse signal from noisy samples, where the sampling is done over some interval $[0, T]$ and the frequencies can be "off-grid". Previous methods for this problem required the gap between frequencies to be ... More

On Communication Complexity of Classification ProblemsNov 16 2017Apr 23 2018This work studies distributed learning in the spirit of Yao's model of communication complexity: consider a two-party setting, where each of the players gets a list of labelled examples and they communicate in order to jointly perform some learning task. ... More

A PRG for Lipschitz Functions of Polynomials with Applications to Sparsest CutNov 06 2012We give improved pseudorandom generators (PRGs) for Lipschitz functions of low-degree polynomials over the hypercube. These are functions of the form psi(P(x)), where P is a low-degree polynomial and psi is a function with small Lipschitz constant. PRGs ... More

An example concerning set addition in F_2^nMar 03 2017Nov 13 2017We construct sets $A, B$ in a vector space over $\mathbb{F}_2$ with the property that $A$ is "statistically" almost closed under addition by $B$ in the sense that $a + b$ almost always lies in $A$ when $a \in A, b \in B$, but which is extremely far from ... More

A bound on partitioning clustersFeb 03 2017May 23 2017Let $X$ be a finite collection of sets (or "clusters"). We consider the problem of counting the number of ways a cluster $A \in X$ can be partitioned into two disjoint clusters $A_1, A_2 \in X$, thus $A = A_1 \uplus A_2$ is the disjoint union of $A_1$ ... More

New results on the least common multiple of consecutive integersAug 11 2008When studying the least common multiple of some finite sequences of integers, the first author introduced the interesting arithmetic functions $g_k$ $(k \in \mathbb{N})$, defined by $g_k(n) := \frac{n (n + 1) ... (n + k)}{\lcm(n, n + 1, >..., n + k)}$ ... More

On the Joint Distribution Of $\mathrm{Sel}_φ(E/\mathbb{Q})$ and $\mathrm{Sel}_{\hatφ}(E^\prime/\mathbb{Q})$ in Quadratic Twist FamiliesFeb 09 2017If $E$ is an elliptic curve with a point of order two, then work of Klagsbrun and Lemke Oliver shows that the distribution of $\dim_{\mathbb{F}_2}\mathrm{Sel}_\phi(E^d/\mathbb{Q}) - \dim_{\mathbb{F}_2} \mathrm{Sel}_{\hat\phi}(E^{\prime d}/\mathbb{Q})$ ... More

Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curvesApr 15 2013Aug 06 2013Using maximal isotropic submodules in a quadratic module over Z_p, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short exact sequences of co-finite type Z_p-modules, and then conjecture that ... More

Best possible densities of Dickson m-tuples, as a consequence of Zhang-Maynard-TaoOct 29 2014Oct 20 2015We determine for what proportion of integers $h$ one now knows that there are infinitely many prime pairs $p,\ p+h$ as a consequence of the Zhang-Maynard-Tao theorem. We consider the natural generalization of this to $k$-tuples of integers, and we determine ... More

Sever: A Robust Meta-Algorithm for Stochastic OptimizationMar 07 2018May 29 2019In high dimensions, most machine learning methods are brittle to even a small fraction of structured outliers. To address this, we introduce a new meta-algorithm that can take in a base learner such as least squares or stochastic gradient descent, and ... More

Sever: A Robust Meta-Algorithm for Stochastic OptimizationMar 07 2018In high dimensions, most machine learning methods are brittle to even a small fraction of structured outliers. To address this, we introduce a new meta-algorithm that can take in a base learner such as least squares or stochastic gradient descent, and ... More

Robustly Learning a Gaussian: Getting Optimal Error, EfficientlyApr 12 2017Nov 05 2017We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise -- where an $\varepsilon$-fraction of our samples were chosen by an adversary. We give robust estimators that achieve estimation error ... More

Being Robust (in High Dimensions) Can Be PracticalMar 02 2017Mar 13 2018Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in theoretical computer ... More

Ergodic Properties of a Class of Discrete Abelian Group Extensions of Rank-One TransformationsMar 28 2008We define a class of discrete abelian group extensions of rank-one transformations and establish necessary and sufficient conditions for these extensions to be power weakly mixing. We show that all members of this class are multiply recurrent. We then ... More

The Optimal Approximation Factor in Density EstimationFeb 10 2019Consider the following problem: given two arbitrary densities $q_1,q_2$ and a sample-access to an unknown target density $p$, find which of the $q_i$'s is closer to $p$ in total variation. A remarkable result due to Yatracos shows that this problem is ... More

Robust polynomial regression up to the information theoretic limitAug 10 2017We consider the problem of robust polynomial regression, where one receives samples $(x_i, y_i)$ that are usually within $\sigma$ of a polynomial $y = p(x)$, but have a $\rho$ chance of being arbitrary adversarial outliers. Previously, it was known how ... More

Pseudorandomness for concentration bounds and signed majoritiesNov 17 2014The problem of constructing pseudorandom generators that fool halfspaces has been studied intensively in recent times. For fooling halfspaces over the hypercube with polynomially small error, the best construction known requires seed-length O(log^2 n) ... More

The Optimal Approximation Factor in Density EstimationFeb 10 2019May 26 2019Consider the following problem: given two arbitrary densities $q_1,q_2$ and a sample-access to an unknown target density $p$, find which of the $q_i$'s is closer to $p$ in total variation. A remarkable result due to Yatracos shows that this problem is ... More

A Stopped Negative Binomial DistributionAug 06 2015Sep 08 2016This paper introduces a new discrete distribution suggested by curtailed sampling rules common in early-stage clinical trials. We derive the distribution of the smallest number of independent Bernoulli(p) trials needed in order to observe either s successes ... More

Robust Learning of Fixed-Structure Bayesian NetworksJun 23 2016We investigate the problem of learning Bayesian networks in an agnostic model where an $\epsilon$-fraction of the samples are adversarially corrupted. Our agnostic learning model is similar to -- in fact, stronger than -- Huber's contamination model in ... More

Pseudorandomness via the discrete Fourier transformJun 14 2015Nov 18 2015We present a new approach to constructing unconditional pseudorandom generators against classes of functions that involve computing a linear function of the inputs. We give an explicit construction of a pseudorandom generator that fools the discrete Fourier ... More

The independence number of the Birkhoff polytope graph, and applications to maximally recoverable codesFeb 19 2017Mar 31 2017Maximally recoverable codes are codes designed for distributed storage which combine quick recovery from single node failure and optimal recovery from catastrophic failure. Gopalan et al [SODA 2017] studied the alphabet size needed for such codes in grid ... More

A Stopped Negative Binomial DistributionAug 06 2015Feb 14 2018This paper introduces a new discrete distribution suggested by curtailed sampling rules common in early-stage clinical trials. We derive the distribution of the smallest number of independent Bernoulli(p) trials needed in order to observe either s successes ... More

Implications of the reported deviations from the standard model for Z->bb and alpha_sOct 24 1995If the reported excess (over the standard model prediction) for Z->bb from LEP persists, and is explained by supersymmetric particles in loops, then we show that (1) a superpartner (chargino and/or stop) will be detected at LEP2, and probably at LEP1.5 ... More

Introduction to "The Supersymmetric World: The Beginnings of The Theory"Feb 24 2001This is the foreword to the book we edited on the origins and early development of supersymmetry, which has been just issued by World Scientific. This book presents a view on the discovery of supersymmetry and pioneering investigations before summer 1976, ... More

A Pseudopolynomial Algorithm for Alexandrov's TheoremDec 30 2008Jan 04 2010Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation ... More

Dirac Semimetals in Two DimensionsApr 29 2015Aug 20 2015Graphene is famous for being a host of 2D Dirac fermions. However, spin-orbit coupling introduces a small gap, so that graphene is formally a quantum spin hall insulator. Here we present symmetry-protected 2D Dirac semimetals, which feature Dirac cones ... More

The Stellar Activity of TRAPPIST-1 and Consequences for the Planetary AtmospheresNov 07 2017The signatures of planets hosted by M dwarfs are more readily detected with transit photometry and radial velocity methods than those of planets around larger stars. Recently, transit photometry was used to discover seven planets orbiting the late-M dwarf ... More

Distinguishing Between Stellar and Planetary Companions With Phase MonitoringMay 25 2012Jul 15 2019Exoplanets which are detected using the radial velocity technique have a well-known ambiguity of their true mass, caused by the unknown inclination of the planetary orbit with respect to the plane of the sky. Constraints on the inclination are aided by ... More

On the Inclination Dependence of Exoplanet Phase SignaturesJan 05 2011Improved photometric sensitivity from space-based telescopes have enabled the detection of phase variations for a small sample of hot Jupiters. However, exoplanets in highly eccentric orbits present unique opportunities to study the effects of drastically ... More

Phase Curves of the Kepler-11 Multi-Planet SystemApr 29 2014The Kepler mission has allowed the detection of numerous multi-planet exosystems where the planetary orbits are relatively compact. The first such system detected was Kepler-11 which has six known planets at the present time. These kinds of systems offer ... More

The Habitable Zone and Extreme Planetary OrbitsMay 11 2012The Habitable Zone for a given star describes the range of circumstellar distances from the star within which a planet could have liquid water on its surface, which depends upon the stellar properties. Here we describe the development of the Habitable ... More

Photometric Phase Variations of Long-Period Eccentric PlanetsSep 24 2010The field of exoplanetary science has diversified rapidly over recent years as the field has progressed from exoplanet detection to exoplanet characterization. For those planets known to transit, the primary transit and secondary eclipse observations ... More

On the Inclination and Habitability of the HD 10180 SystemAug 18 2014There are numerous multi-planet systems that have now been detected via a variety of techniques. These systems exhibit a range of both planetary properties and orbital configurations. For those systems without detected planetary transits, a significant ... More

A new (string motivated) approach to the little hierarchy problemMay 18 2011May 28 2011We point out that in theories where the gravitino mass, $M_{3/2}$, is in the range (10-50)TeV, with soft-breaking scalar masses and trilinear couplings of the same order, there exists a robust region of parameter space where the conditions for electroweak ... More

Decoupling Phase Variations in Multi-Planet SystemsNov 28 2012Due to the exquisite photometric precision, transiting exoplanet discoveries from the Kepler mission are enabling several new techniques of confirmation and characterization. One of these newly accessible techniques analyzes the phase variations of planets ... More

Reality as a simulation of reality: robot illusions, fundamental limits, and a physical demonstrationSep 09 2019We consider problems in which robots conspire to present a view of the world that differs from reality. The inquiry is motivated by the problem of validating robot behavior physically despite there being a discrepancy between the robots we have at hand ... More

Testing Bayesian NetworksDec 09 2016This work initiates a systematic investigation of testing {\em high-dimensional} structured distributions by focusing on testing {\em Bayesian networks} -- the prototypical family of directed graphical models. A Bayesian network is defined by a directed ... More

The entropy of lies: playing twenty questions with a liarNov 06 2018`Twenty questions' is a guessing game played by two players: Bob thinks of an integer between $1$ and $n$, and Alice's goal is to recover it using a minimal number of Yes/No questions. Shannon's entropy has a natural interpretation in this context. It ... More

Weighing the universe with accelerators and detectorsMay 16 2000Aug 02 2000Suppose the lightest superpartner (LSP) is observed at colliders, and WIMPs are detected in explicit experiments. We point out that one cannot immediately conclude that cold dark matter (CDM) of the universe has been observed, and we determine what measurements ... More