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Small Approximate Pareto Sets for Bi-objective Shortest Paths and Other ProblemsMay 17 2008We investigate the problem of computing a minimum set of solutions that approximates within a specified accuracy $\epsilon$ the Pareto curve of a multiobjective optimization problem. We show that for a broad class of bi-objective problems (containing ... More

On the Complexity of the Inverse Semivalue Problem for Weighted Voting GamesDec 31 2018Weighted voting games are a family of cooperative games, typically used to model voting situations where a number of agents (players) vote against or for a proposal. In such games, a proposal is accepted if an appropriately weighted sum of the votes exceeds ... 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

Improved Approximation of Linear Threshold FunctionsOct 19 2009We prove two main results on how arbitrary linear threshold functions $f(x) = \sign(w\cdot x - \theta)$ over the $n$-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every $n$-variable threshold ... 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

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

The Inverse Shapley Value ProblemDec 20 2012For $f$ a weighted voting scheme used by $n$ voters to choose between two candidates, the $n$ \emph{Shapley-Shubik Indices} (or {\em Shapley values}) of $f$ provide a measure of how much control each voter can exert over the overall outcome of the vote. ... More

Efficient Algorithms and Lower Bounds for Robust Linear RegressionMay 31 2018We study the problem of high-dimensional linear regression in a robust model where an $\epsilon$-fraction of the samples can be adversarially corrupted. We focus on the fundamental setting where the covariates of the uncorrupted samples are drawn from ... More

Fourier-Based Testing for Families of DistributionsJun 18 2017Aug 08 2017We study the general problem of testing whether an unknown distribution belongs to a specified family of distributions. More specifically, given a distribution family $\mathcal{P}$ and sample access to an unknown discrete distribution $\mathbf{P}$, we ... 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

Distribution-Independent PAC Learning of Halfspaces with Massart NoiseJun 24 2019We study the problem of {\em distribution-independent} PAC learning of halfspaces in the presence of Massart noise. Specifically, we are given a set of labeled examples $(\mathbf{x}, y)$ drawn from a distribution $\mathcal{D}$ on $\mathbb{R}^{d+1}$ such ... 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

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

Learning $k$-Modal Distributions via TestingJul 13 2011Sep 14 2014A $k$-modal probability distribution over the discrete domain $\{1,...,n\}$ is one whose histogram has at most $k$ "peaks" and "valleys." Such distributions are natural generalizations of monotone ($k=0$) and unimodal ($k=1$) probability distributions, ... 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

A robust Khintchine inequality, and algorithms for computing optimal constants in Fourier analysis and high-dimensional geometryJul 10 2012May 03 2013This paper makes two contributions towards determining some well-studied optimal constants in Fourier analysis \newa{of Boolean functions} and high-dimensional geometry. \begin{enumerate} \item It has been known since 1994 \cite{GL:94} that every linear ... 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

High-Dimensional Robust Mean Estimation in Nearly-Linear TimeNov 23 2018We study the fundamental problem of high-dimensional mean estimation in a robust model where a constant fraction of the samples are adversarially corrupted. Recent work gave the first polynomial time algorithms for this problem with dimension-independent ... 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

Deterministic Approximate Counting for Juntas of Degree-$2$ Polynomial Threshold FunctionsNov 27 2013Let $g: \{-1,1\}^k \to \{-1,1\}$ be any Boolean function and $q_1,\dots,q_k$ be any degree-2 polynomials over $\{-1,1\}^n.$ We give a \emph{deterministic} algorithm which, given as input explicit descriptions of $g,q_1,\dots,q_k$ and an accuracy parameter ... More

Inverse problems in approximate uniform generationNov 07 2012We initiate the study of \emph{inverse} problems in approximate uniform generation, focusing on uniform generation of satisfying assignments of various types of Boolean functions. In such an inverse problem, the algorithm is given uniform random satisfying ... 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

A Polynomial Time Algorithm for Maximum Likelihood Estimation of Multivariate Log-concave DensitiesDec 13 2018We study the problem of computing the maximum likelihood estimator (MLE) of multivariate log-concave densities. Our main result is the first computationally efficient algorithm for this problem. In more detail, we give an algorithm that, on input a set ... 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

How good is the Chord algorithm?Sep 26 2013The Chord algorithm is a popular, simple method for the succinct approximation of curves, which is widely used, under different names, in a variety of areas, such as, multiobjective and parametric optimization, computational geometry, and graphics. We ... More

Learning transformed product distributionsMar 03 2011We consider the problem of learning an unknown product distribution $X$ over $\{0,1\}^n$ using samples $f(X)$ where $f$ is a \emph{known} transformation function. Each choice of a transformation function $f$ specifies a learning problem in this framework. ... More

Learning Poisson Binomial DistributionsJul 13 2011Feb 17 2015We consider a basic problem in unsupervised learning: learning an unknown \emph{Poisson Binomial Distribution}. A Poisson Binomial Distribution (PBD) over $\{0,1,\dots,n\}$ is the distribution of a sum of $n$ independent Bernoulli random variables which ... More

Deterministic Approximate Counting for Degree-$2$ Polynomial Threshold FunctionsNov 27 2013We give a {\em deterministic} algorithm for approximately computing the fraction of Boolean assignments that satisfy a degree-$2$ polynomial threshold function. Given a degree-2 input polynomial $p(x_1,\dots,x_n)$ and a parameter $\eps > 0$, the algorithm ... More

Fast and Sample Near-Optimal Algorithms for Learning Multidimensional HistogramsFeb 23 2018We study the problem of robustly learning multi-dimensional histograms. A $d$-dimensional function $h: D \rightarrow \mathbb{R}$ is called a $k$-histogram if there exists a partition of the domain $D \subseteq \mathbb{R}^d$ into $k$ axis-aligned rectangles ... More

Differentially Private Identity and Closeness Testing of Discrete DistributionsJul 18 2017We investigate the problems of identity and closeness testing over a discrete population from random samples. Our goal is to develop efficient testers while guaranteeing Differential Privacy to the individuals of the population. We describe an approach ... More

Playing Anonymous Games using Simple StrategiesAug 25 2016We investigate the complexity of computing approximate Nash equilibria in anonymous games. Our main algorithmic result is the following: For any $n$-player anonymous game with a bounded number of strategies and any constant $\delta>0$, an $O(1/n^{1-\delta})$-approximate ... More

Efficiency-Revenue Trade-offs in AuctionsMay 14 2012When agents with independent priors bid for a single item, Myerson's optimal auction maximizes expected revenue, whereas Vickrey's second-price auction optimizes social welfare. We address the natural question of trade-offs between the two criteria, that ... More

Collision-based Testers are Optimal for Uniformity and ClosenessNov 11 2016We study the fundamental problems of (i) uniformity testing of a discrete distribution, and (ii) closeness testing between two discrete distributions with bounded $\ell_2$-norm. These problems have been extensively studied in distribution testing and ... More

Faster Algorithms for High-Dimensional Robust Covariance EstimationJun 11 2019We study the problem of estimating the covariance matrix of a high-dimensional distribution when a small constant fraction of the samples can be arbitrarily corrupted. Recent work gave the first polynomial time algorithms for this problem with near-optimal ... More

The Complexity of Optimal Multidimensional PricingNov 09 2013We resolve the complexity of revenue-optimal deterministic auctions in the unit-demand single-buyer Bayesian setting, i.e., the optimal item pricing problem, when the buyer's values for the items are independent. We show that the problem of computing ... More

Equipping Experts/Bandits with Long-term MemoryMay 30 2019We propose the first reduction-based approach to obtaining long-term memory guarantees for online learning in the sense of Bousquet and Warmuth, 2002, by reducing the problem to achieving typical switching regret. Specifically, for the classical expert ... More

Testing Shape Restrictions of Discrete DistributionsJul 13 2015Jan 21 2016We study the question of testing structured properties (classes) of discrete distributions. Specifically, given sample access to an arbitrary distribution $D$ over $[n]$ and a property $\mathcal{P}$, the goal is to distinguish between $D\in\mathcal{P}$ ... More

Hardness Results for Agnostically Learning Low-Degree Polynomial Threshold FunctionsOct 18 2010Hardness results for maximum agreement problems have close connections to hardness results for proper learning in computational learning theory. In this paper we prove two hardness results for the problem of finding a low degree polynomial threshold function ... More

Nearly optimal solutions for the Chow Parameters Problem and low-weight approximation of halfspacesJun 05 2012The \emph{Chow parameters} of a Boolean function $f: \{-1,1\}^n \to \{-1,1\}$ are its $n+1$ degree-0 and degree-1 Fourier coefficients. It has been known since 1961 (Chow, Tannenbaum) that the (exact values of the) Chow parameters of any linear threshold ... More

Near-Optimal Density Estimation in Near-Linear Time Using Variable-Width HistogramsNov 01 2014Let $p$ be an unknown and arbitrary probability distribution over $[0,1)$. We consider the problem of {\em density estimation}, in which a learning algorithm is given i.i.d. draws from $p$ and must (with high probability) output a hypothesis distribution ... More

Efficient Density Estimation via Piecewise Polynomial ApproximationMay 14 2013We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let $p$ be an arbitrary distribution over an interval $I$ which is $\tau$-close ... More

Learning mixtures of structured distributions over discrete domainsOct 02 2012Let $\mathfrak{C}$ be a class of probability distributions over the discrete domain $[n] = \{1,...,n\}.$ We show that if $\mathfrak{C}$ satisfies a rather general condition -- essentially, that each distribution in $\mathfrak{C}$ can be well-approximated ... More

Average sensitivity and noise sensitivity of polynomial threshold functionsSep 28 2009Oct 19 2009We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of degree-$d$ polynomial threshold functions (PTFs). These bounds hold both for PTFs over the Boolean hypercube and for PTFs over $\R^n$ under the standard $n$-dimensional ... More

A regularity lemma, and low-weight approximators, for low-degree polynomial threshold functionsSep 25 2009May 06 2010We give a "regularity lemma" for degree-d polynomial threshold functions (PTFs) over the Boolean cube {-1,1}^n. This result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions ... 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

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

Sample-Optimal Density Estimation in Nearly-Linear TimeJun 01 2015We design a new, fast algorithm for agnostically learning univariate probability distributions whose densities are well approximated by piecewise polynomial functions. Let $f$ be the density function of an arbitrary univariate distribution, and suppose ... More

A Polynomial Time Algorithm for Log-Concave Maximum Likelihood via Locally Exponential FamiliesJul 18 2019We consider the problem of computing the maximum likelihood multivariate log-concave distribution for a set of points. Specifically, we present an algorithm which, given $n$ points in $\mathbb{R}^d$ and an accuracy parameter $\epsilon>0$, runs in time ... More

Lower Bounds for Parallel and Randomized Convex OptimizationNov 05 2018Jan 23 2019We study the question of whether parallelization in the exploration of the feasible set can be used to speed up convex optimization, in the local oracle model of computation. We show that the answer is negative for both deterministic and randomized algorithms ... More

Near-Optimal Sample Complexity Bounds for Maximum Likelihood Estimation of Multivariate Log-concave DensitiesFeb 28 2018Dec 05 2018We study the problem of learning multivariate log-concave densities with respect to a global loss function. We obtain the first upper bound on the sample complexity of the maximum likelihood estimator (MLE) for a log-concave density on $\mathbb{R}^d$, ... 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

Optimal Identity Testing with High ProbabilityAug 09 2017Jan 15 2019We study the problem of testing identity against a given distribution with a focus on the high confidence regime. More precisely, given samples from an unknown distribution $p$ over $n$ elements, an explicitly given distribution $q$, and parameters $0< ... More

Bounded Independence Fools HalfspacesFeb 21 2009We show that any distribution on {-1,1}^n that is k-wise independent fools any halfspace h with error \eps for k = O(\log^2(1/\eps) /\eps^2). Up to logarithmic factors, our result matches a lower bound by Benjamini, Gurel-Gurevich, and Peled (2007) showing ... More

Optimal Algorithms for Testing Closeness of Discrete DistributionsAug 19 2013We study the question of closeness testing for two discrete distributions. More precisely, given samples from two distributions $p$ and $q$ over an $n$-element set, we wish to distinguish whether $p=q$ versus $p$ is at least $\eps$-far from $q$, in either ... More

Fast Algorithms for Segmented RegressionJul 14 2016We study the fixed design segmented regression problem: Given noisy samples from a piecewise linear function $f$, we want to recover $f$ up to a desired accuracy in mean-squared error. Previous rigorous approaches for this problem rely on dynamic programming ... More

Robust Learning of Fixed-Structure Bayesian NetworksJun 23 2016Oct 29 2018We investigate the problem of learning Bayesian networks in a robust model where an $\epsilon$-fraction of the samples are adversarially corrupted. In this work, we study the fully observable discrete case where the structure of the network is given. ... More

Solving Packing and Covering LPs in $\tilde{O}(\frac{1}{ε^2})$ Distributed Iterations with a Single Algorithm and Simpler AnalysisOct 24 2017Packing and covering linear programs belong to the narrow class of linear programs that are efficiently solvable in parallel and distributed models of computation, yet are a powerful modeling tool for a wide range of fundamental problems in theoretical ... More

Accelerated Extra-Gradient Descent: A Novel Accelerated First-Order MethodJun 14 2017Feb 10 2018We provide a novel accelerated first-order method that achieves the asymptotically optimal convergence rate for smooth functions in the first-order oracle model. To this day, Nesterov's Accelerated Gradient Descent (AGD) and variations thereof were the ... More

Generalized Momentum-Based Methods: A Hamiltonian PerspectiveJun 02 2019We take a Hamiltonian-based perspective to generalize Nesterov's accelerated gradient descent and Polyak's heavy ball method to a broad class of momentum methods in the setting of (possibly) constrained minimization in Banach spaces. Our perspective leads ... 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

Torus Fibrations of Calabi-Yau Hypersurfaces in Toric Varieties and Mirror SymmetryJun 17 1998We consider regular Calabi-Yau hypersurfaces in $N$-dimensional smooth toric varieties. On such a hypersurface in the neighborhood of the large complex structure limit point we construct a fibration over a sphere $S^{N-1}$ whose generic fibers are tori ... More

Remark on the paper "Eigenvalues of the Witten-Laplacian on compact Riemannian manifolds" Q.-M. Cheng and L. Zeng [arXiv:1304.3212]Apr 13 2013Why the results of that article [arXiv:1304.3212] are immediate consequences of known ones.

On Cohomology and vector bundles over monoid schmesAug 23 2013Feb 12 2014The aim of this paper is to study the cohomology theory of monoid schemes in general and apply it to vector and line bundles. We will prove that over separated monoid schemes, any vector bundle is a coproduct of line bundles and then go on to study the ... More

Dicrete Analogues of the Laguerre InequalityApr 09 2002It is shown that $\sum_{j=-m}^m (-1)^j \frac{f(x-j)(f(x+j)}{(m-j)! (m+j)!} \ge 0,$ $m=0,1,...,$ where $f(x)$ is either a real polynomial with only real zeros or an allied entire function of a special type, provided the distance between two consecutive ... More

Harmonic functions on the Sierpinski triangleMay 20 2012In this paper, we give a few results on the local behavior of harmonic functions on the Sierpinski triangle - more precisely, of their restriction to a side of the triangle. First we present a general formula that gives the H\"older exponent of such a ... More

Construction of Milnorian representationsFeb 20 2018Jun 28 2019We prove a partial converse to the main theorem of the author's previous paper "Proper affine actions: a sufficient criterion" (submitted; available at arXiv:1612.08942). More precisely, let $G$ be a semisimple real Lie group with a representation $\rho$ ... More

Proper affine actions on semisimple Lie algebrasJun 23 2014May 12 2016For any noncompact semisimple real Lie group $G$, we construct a group of affine transformations of its Lie algebra $\mathfrak{g}$ whose linear part is Zariski-dense in $\operatorname{Ad} G$ and which is free, nonabelian and acts properly discontinuously ... More

On the Gravitational Wave Background from Black Hole Binaries after the First LIGO DetectionsSep 12 2016The detection of gravitational waves from the merger of binary black holes by the LIGO Collaboration has opened a new window to astrophysics. With the sensitivities of ground based detectors in the coming years we can only detect the local black hole ... More

Optimizing the Ptolemaic Model of Planetary and Solar MotionFeb 06 2015Mar 10 2015Ptolemy-s planetary model is an ancient geocentric astronomical model, describing the observed motion of the Sun and the planets. Ptolemy accounted for the deviations of planetary orbits from perfect circles by introducing two small and equal shifts into ... More

Some asymptotics for the Bessel functions with an explicit error termJul 11 2011Jul 14 2011We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel function ... More

Critical collapse and black hole formation within an expanding perfect fluidJun 27 2011Oct 20 2016Following on after three previous papers discussing the formation of primordial black holes in the early universe during the radiation dominated era, we present here related results considering the theoretical possibility of having a fluid with a different ... More

$C$ is not equivalent to $C^-$ in its Jacobian: a tropical point of viewMay 20 2013Oct 08 2013We show that the Abel-Jacobi image of a tropical curve $C$ in its Jacobian $J(C)$ is not algebraically equivalent to its reflection by using a simple calculation in tropical homology.

On the equvialence of colimits and 2-colimitsMay 27 2019We compare the colimit and 2-colimit of strict 2-functors in the 2-category of groupoids, over a certain type of posets. These posets are of special importance, as they correspond to coverings of a topological space. The main result of this paper gives ... More

Turán Inequalities for Three Term Recurrences with Monotonic CoefficientsJan 17 2011We establish some new Tur\'an's type inequalities for orthogonal polynomials defined by a three-term recurrence with monotonic coefficients. As a corollary we deduce asymptotic bounds on the extreme zeros of orthogonal polynomials with polynomially growing ... More

Multiplicity of zeros and discrete orthogonal polynomialsApr 25 2002We consider a problem of bounding the maximal possible multiplicity of a zero at of some expansions $\sum a_i F_i(x)$, at a certain point $c,$ depending on the chosen family $\{F_i \}$. The most important example is a polynomial with $c=1.$ It is shown ... More

On the equvialence of colimits and 2-colimitsMay 27 2019Jul 20 2019We compare the colimit and 2-colimit of strict 2-functors in the 2-category of groupoids, over a certain type of posets. These posets are of special importance, as they correspond to coverings of a topological space. The main result of this paper gives ... More

Generalized Momentum-Based Methods: A Hamiltonian PerspectiveJun 02 2019Jun 23 2019We take a Hamiltonian-based perspective to generalize Nesterov's accelerated gradient descent and Polyak's heavy ball method to a broad class of momentum methods in the setting of (possibly) constrained minimization in Banach spaces. Our perspective leads ... More

A Polynomial-time Approximation Scheme for Fault-tolerant Distributed StorageJul 13 2013We consider a problem which has received considerable attention in systems literature because of its applications to routing in delay tolerant networks and replica placement in distributed storage systems. In abstract terms the problem can be stated as ... More

On the Distribution of the Fourier Spectrum of HalfspacesFeb 29 2012Bourgain showed that any noise stable Boolean function $f$ can be well-approximated by a junta. In this note we give an exponential sharpening of the parameters of Bourgain's result under the additional assumption that $f$ is a halfspace.

On approximation of ultraspherical polynomials in the oscillatory regionAug 29 2016For $k \ge 2$ even, and $ \alpha \ge -(2k+1)/4 $, we provide a uniform approximation of the ultraspherical polynomials $ P_k^{(\alpha,\, \alpha)}(x) $ in the oscillatory region with a very explicit error term. In fact, our result covers all $\alpha$ for ... More

Tropical theta characteristicsDec 19 2007Feb 18 2009This note is a follow up of math.AG/0612267v2 and it is largely inspired by a beautiful description of Baker-Norine of non-effective degree (g-1) divisors via chip-firing game. We consider the set of all theta characteristics on a tropical curve and identify ... More

The Orlik-Solomon Algebra and the Bergman Fan of a MatroidSep 07 2012Oct 08 2013Given a matroid $M$ one can define its Orlik-Solomon algebra $OS(M)$ and the Bergman fan $\Sigma_0(M)$. On the other hand to any rational polyhedral fan $\Sigma$ one can associate its tropical homology and cohomology groups $\F_\bullet(\Sigma)$, $\F^\bullet ... More

Theta-functions for indefinite polarizationsNov 16 2000Jul 10 2003We propose a generalization of the classical theta function to higher cohomology of the polarization line bundle on a family of complex tori with positive index. The constructed cocycles vary horizontally with respect to the (projective) flat connection ... More

Proper affine actions: a sufficient criterionDec 28 2016Oct 19 2018For a semisimple real Lie group $G$ with an irreducible representation $\rho$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $\rho$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense ... More

New bounds on the Hermite polynomialsJan 23 2004We shall establish two-side explicit inequalities, which are asymptotically sharp up to a constant factor, on the maximum value of $|H_k(x)| e^{-x^2/2},$ on the real axis, where $H_k$ are the Hermite polynomials.

Turan inequalities and zeros of orthogonal polynomialsJan 22 2004We use Turan type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the three term ... More

Universal Padé approximants and their behaviour on the boundaryOct 05 2013There are several kinds of universal Taylor series. In one such kind the universal approximation is required at every boundary point of the domain of definition $\OO$ of the universal function $f$. In another kind the universal approximation is not required ... More

A translation of Y. Benoist's "Actions propres sur les espaces homogènes réductifs"Feb 20 2018This is a translation of Yves Benoist's "Actions propres sur les espaces homog\`enes r\'eductifs", Ann. of Math., 144:315-347, 1996.

Proper affine actions in non-swinging representationsMay 12 2016Sep 20 2018For a semisimple real Lie group $G$ with an irreducible representation $\rho$ on a finite-dimensional real vector space $V$, we give a sufficient criterion on $\rho$ for existence of a group of affine transformations of $V$ whose linear part is Zariski-dense ... More

On the group of separable quadratic algebras and stacksApr 09 2016Sep 18 2016The aim of this paper is to study the group of isomorphism classes of torsors of finite flat group schemes of rank 2 over a commutative ring $R$. This, in particular, generalises the group of quadratic algebras (free or projective), which is especially ... More

Robust Estimators in High Dimensions without the Computational IntractabilityApr 21 2016We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an $\varepsilon$-fraction of the samples. Such questions have a rich history spanning statistics, machine learning and theoretical ... More