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Statistical Limits of Convex RelaxationsMar 04 2015Mar 14 2015Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this paper, we study ... More

Sample Efficient Stochastic Variance-Reduced Cubic Regularization MethodNov 29 2018We propose a sample efficient stochastic variance-reduced cubic regularization (Lite-SVRC) algorithm for finding the local minimum efficiently in nonconvex optimization. The proposed algorithm achieves a lower sample complexity of Hessian matrix computation ... More

On the Convergence of Adaptive Gradient Methods for Nonconvex OptimizationAug 16 2018Nov 27 2018Adaptive gradient methods are workhorses in deep learning. However, the convergence guarantees of adaptive gradient methods for nonconvex optimization have not been sufficiently studied. In this paper, we provide a sharp analysis of a recently proposed ... More

A Generalization Theory of Gradient Descent for Learning Over-parameterized Deep ReLU NetworksFeb 04 2019Feb 15 2019Empirical studies show that gradient based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. While a line of recent ... More

Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural NetworksMay 30 2019Jun 06 2019We study the training and generalization of deep neural networks (DNNs) in the over-parameterized regime, where the network width (i.e., number of hidden nodes per layer) is much larger than the number of training data points. We show that, the expected ... More

Communication-efficient Distributed Sparse Linear Discriminant AnalysisOct 15 2016We propose a communication-efficient distributed estimation method for sparse linear discriminant analysis (LDA) in the high dimensional regime. Our method distributes the data of size $N$ into $m$ machines, and estimates a local sparse LDA estimator ... More

Closing the Generalization Gap of Adaptive Gradient Methods in Training Deep Neural NetworksJun 18 2018Adaptive gradient methods, which adopt historical gradient information to automatically adjust the learning rate, have been observed to generalize worse than stochastic gradient descent (SGD) with momentum in training deep neural networks. This leaves ... More

Towards Faster Rates and Oracle Property for Low-Rank Matrix EstimationMay 18 2015Jul 06 2015We present a unified framework for low-rank matrix estimation with nonconvex penalties. We first prove that the proposed estimator attains a faster statistical rate than the traditional low-rank matrix estimator with nuclear norm penalty. Moreover, we ... More

Generalization Error Bounds of Gradient Descent for Learning Over-parameterized Deep ReLU NetworksFeb 04 2019Apr 02 2019Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. While a line of recent ... More

Lower Bounds for Smooth Nonconvex Finite-Sum OptimizationJan 31 2019Smooth finite-sum optimization has been widely studied in both convex and nonconvex settings. However, existing lower bounds for finite-sum optimization are mostly limited to the setting where each component function is (strongly) convex, while the lower ... More

DP-LSSGD: A Stochastic Optimization Method to Lift the Utility in Privacy-Preserving ERMJun 28 2019Machine learning (ML) models trained by differentially private stochastic gradient descent (DP-SGD) has much lower utility than the non-private ones. To mitigate this degradation, we propose a DP Laplacian smoothing SGD (DP-LSSGD) for privacy-preserving ... More

Generalization Bounds of Stochastic Gradient Descent for Wide and Deep Neural NetworksMay 30 2019We study the training and generalization of deep neural networks (DNNs) in the over-parameterized regime, where the network width (i.e., number of hidden nodes per layer) is much larger than the number of training data points. We show that, the expected ... More

High Dimensional Multivariate Regression and Precision Matrix Estimation via Nonconvex OptimizationJun 02 2016We propose a nonconvex estimator for joint multivariate regression and precision matrix estimation in the high dimensional regime, under sparsity constraints. A gradient descent algorithm with hard thresholding is developed to solve the nonconvex estimator, ... More

An Improved Analysis of Training Over-parameterized Deep Neural NetworksJun 11 2019A recent line of research has shown that gradient-based algorithms with random initialization can converge to the global minima of the training loss for over-parameterized (i.e., sufficiently wide) deep neural networks. However, the condition on the width ... More

Stochastic Recursive Variance-Reduced Cubic Regularization MethodsJan 31 2019Stochastic Variance-Reduced Cubic regularization (SVRC) algorithms have received increasing attention due to its improved gradient/Hessian complexities (i.e., number of queries to stochastic gradient/Hessian oracles) to find local minima for nonconvex ... More

Fast and Sample Efficient Inductive Matrix Completion via Multi-Phase Procrustes FlowMar 03 2018We revisit the inductive matrix completion problem that aims to recover a rank-$r$ matrix with ambient dimension $d$ given $n$ features as the side prior information. The goal is to make use of the known $n$ features to reduce sample and computational ... More

A Universal Variance Reduction-Based Catalyst for Nonconvex Low-Rank Matrix RecoveryJan 09 2017Jan 19 2017We propose a generic framework based on a new stochastic variance-reduced gradient descent algorithm for accelerating nonconvex low-rank matrix recovery. Starting from an appropriate initial estimator, our proposed algorithm performs projected gradient ... More

Sharp Computational-Statistical Phase Transitions via Oracle Computational ModelDec 30 2015We study the fundamental tradeoffs between computational tractability and statistical accuracy for a general family of hypothesis testing problems with combinatorial structures. Based upon an oracle model of computation, which captures the interactions ... More

Stochastic Variance-Reduced Hamilton Monte Carlo MethodsFeb 13 2018We propose a fast stochastic Hamilton Monte Carlo (HMC) method, for sampling from a smooth and strongly log-concave distribution. At the core of our proposed method is a variance reduction technique inspired by the recent advance in stochastic optimization. ... More

An Improved Convergence Analysis of Stochastic Variance-Reduced Policy GradientMay 29 2019We revisit the stochastic variance-reduced policy gradient (SVRPG) method proposed by Papini et al. (2018) for reinforcement learning. We provide an improved convergence analysis of SVRPG and show that it can find an $\epsilon$-approximate stationary ... More

Stochastic Nested Variance Reduction for Nonconvex OptimizationJun 20 2018We study finite-sum nonconvex optimization problems, where the objective function is an average of $n$ nonconvex functions. We propose a new stochastic gradient descent algorithm based on nested variance reduction. Compared with conventional stochastic ... More

Third-order Smoothness Helps: Even Faster Stochastic Optimization Algorithms for Finding Local MinimaDec 18 2017We propose stochastic optimization algorithms that can find local minima faster than existing algorithms for nonconvex optimization problems, by exploiting the third-order smoothness to escape non-degenerate saddle points more efficiently. More specifically, ... More

Generalized Fisher Score for Feature SelectionFeb 14 2012Fisher score is one of the most widely used supervised feature selection methods. However, it selects each feature independently according to their scores under the Fisher criterion, which leads to a suboptimal subset of features. In this paper, we present ... More

A Unified Framework for Low-Rank plus Sparse Matrix RecoveryFeb 21 2017Feb 20 2018We propose a unified framework to solve general low-rank plus sparse matrix recovery problems based on matrix factorization, which covers a broad family of objective functions satisfying the restricted strong convexity and smoothness conditions. Based ... More

Stochastic Variance-reduced Gradient Descent for Low-rank Matrix Recovery from Linear MeasurementsJan 02 2017Jan 16 2017We study the problem of estimating low-rank matrices from linear measurements (a.k.a., matrix sensing) through nonconvex optimization. We propose an efficient stochastic variance reduced gradient descent algorithm to solve a nonconvex optimization problem ... More

A Unified Computational and Statistical Framework for Nonconvex Low-Rank Matrix EstimationOct 17 2016We propose a unified framework for estimating low-rank matrices through nonconvex optimization based on gradient descent algorithm. Our framework is quite general and can be applied to both noisy and noiseless observations. In the general case with noisy ... More

A Frank-Wolfe Framework for Efficient and Effective Adversarial AttacksNov 27 2018Depending on how much information an adversary can access to, adversarial attacks can be classified as white-box attack and black-box attack. In both cases, optimization-based attack algorithms can achieve relatively low distortions and high attack success ... More

Communication-efficient Distributed Estimation and Inference for Transelliptical Graphical ModelsDec 29 2016We propose communication-efficient distributed estimation and inference methods for the transelliptical graphical model, a semiparametric extension of the elliptical distribution in the high dimensional regime. In detail, the proposed method distributes ... More

Speeding Up Latent Variable Gaussian Graphical Model Estimation via Nonconvex OptimizationsFeb 28 2017We study the estimation of the latent variable Gaussian graphical model (LVGGM), where the precision matrix is the superposition of a sparse matrix and a low-rank matrix. In order to speed up the estimation of the sparse plus low-rank components, we propose ... More

Ambiguous Proximity DistributionJun 02 2014Proximity Distribution Kernel is an effective method for bag-of-featues based image representation. In this paper, we investigate the soft assignment of visual words to image features for proximity distribution. Visual word contribution function is proposed ... More

Saving Gradient and Negative Curvature Computations: Finding Local Minima More EfficientlyDec 11 2017We propose a family of nonconvex optimization algorithms that are able to save gradient and negative curvature computations to a large extent, and are guaranteed to find an approximate local minimum with improved runtime complexity. At the core of our ... More

Finding Local Minima via Stochastic Nested Variance ReductionJun 22 2018We propose two algorithms that can find local minima faster than the state-of-the-art algorithms in both finite-sum and general stochastic nonconvex optimization. At the core of the proposed algorithms is $\text{One-epoch-SNVRG}^+$ using stochastic nested ... More

Stochastic Variance-Reduced Cubic Regularized Newton MethodFeb 13 2018We propose a stochastic variance-reduced cubic regularized Newton method for non-convex optimization. At the core of our algorithm is a novel semi-stochastic gradient along with a semi-stochastic Hessian, which are specifically designed for cubic regularization ... More

High Dimensional Expectation-Maximization Algorithm: Statistical Optimization and Asymptotic NormalityDec 30 2014Jan 27 2015We provide a general theory of the expectation-maximization (EM) algorithm for inferring high dimensional latent variable models. In particular, we make two contributions: (i) For parameter estimation, we propose a novel high dimensional EM algorithm ... More

Learning One-hidden-layer ReLU Networks via Gradient DescentJun 20 2018We study the problem of learning one-hidden-layer neural networks with Rectified Linear Unit (ReLU) activation function, where the inputs are sampled from standard Gaussian distribution and the outputs are generated from a noisy teacher network. We analyze ... More

Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU NetworksNov 21 2018Dec 27 2018We study the problem of training deep neural networks with Rectified Linear Unit (ReLU) activation function using gradient descent and stochastic gradient descent. In particular, we study the binary classification problem and show that for a broad family ... More

Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex OptimizationJul 20 2017Feb 19 2018We present a unified framework to analyze the global convergence of Langevin dynamics based algorithms for nonconvex finite-sum optimization with $n$ component functions. At the core of our analysis is a direct analysis of the ergodicity of the numerical ... More

Local and Global Inference for High Dimensional Nonparanormal Graphical ModelsFeb 09 2015Jun 30 2015This paper proposes a unified framework to quantify local and global inferential uncertainty for high dimensional nonparanormal graphical models. In particular, we consider the problems of testing the presence of a single edge and constructing a uniform ... More

Robust Wirtinger Flow for Phase Retrieval with Arbitrary CorruptionApr 20 2017Jan 03 2018We consider the robust phase retrieval problem of recovering the unknown signal from the magnitude-only measurements, where the measurements can be contaminated by both sparse arbitrary corruption and bounded random noise. We propose a new nonconvex algorithm ... More

Grid R-CNN Plus: Faster and BetterJun 13 2019Grid R-CNN is a well-performed objection detection framework. It transforms the traditional box offset regression problem into a grid point estimation problem. With the guidance of the grid points, it can obtain high-quality localization results. However, ... More

Combining RGB and Points to Predict Grasping Region for Robotic Bin-PickingApr 16 2019This paper focuses on a robotic picking tasks in cluttered scenario. Because of the diversity of objects and clutter by placing, it is much difficult to recognize and estimate their pose before grasping. Here, we use U-net, a special Convolution Neural ... More

Suction Grasp Region Prediction using Self-supervised Learning for Object Picking in Dense ClutterApr 16 2019This paper focuses on robotic picking tasks in cluttered scenario. Because of the diversity of poses, types of stack and complicated background in bin picking situation, it is much difficult to recognize and estimate their pose before grasping them. Here, ... More

Combining RGB and Points to Predict Grasping Region for Robotic Bin-PickingApr 16 2019Apr 24 2019This paper focuses on a robotic picking tasks in cluttered scenario. Because of the diversity of objects and clutter by placing, it is much difficult to recognize and estimate their pose before grasping. Here, we use U-net, a special Convolution Neural ... More

Channel Estimation for Opportunistic Spectrum Access: Uniform and Random SensingMay 14 2010The knowledge of channel statistics can be very helpful in making sound opportunistic spectrum access decisions. It is therefore desirable to be able to efficiently and accurately estimate channel statistics. In this paper we study the problem of optimally ... More

Decomposition Algorithm for Median Graph of Triangulation of a Bordered 2D SurfaceJun 22 2010Jul 12 2010This paper develops an algorithm that identifies and decomposes a median graph of a triangulation of a 2-dimensional (2D) oriented bordered surface and in addition restores all corresponding triangulation whenever they exist. The algorithm is based on ... More

A central limit theorem for fluctuations in one dimensional stochastic homogenizationAug 20 2015In this paper, we analyze the random fluctuations in a one dimensional stochastic homogenization problem and prove a central limit result, i.e., the first order fluctuations can be described by a Gaussian process that solves an SPDE with additive spatial ... More

Covering of sparse subgraphs and Packing of rigid subgraphsMay 01 2014May 21 2014A graph $G$ is sparse if for every $X\subseteq V(G)$ with $|X|\ge 2$, the number of edges induced by $X$ is at most $2|X|-3$. We prove a characterization of graphs that can be decomposed to sparse subgraphs, which extends the well-known result of Nash-Williams ... More

Entangled games do not require much entanglement (withdrawn)Aug 24 2009Sep 03 2009We prove an explicit upper bound on the amount of entanglement required by any strategy in a two-player cooperative game with classical questions and quantum answers. Specifically, we show that every strategy for a game with n-bit questions and n-qubit ... More

Properties of Local Quantum Operations with Shared EntanglementMay 15 2008Jul 28 2009Multi-party local quantum operations with shared quantum entanglement or shared classical randomness are studied. The following facts are established: (i) There is a ball of local operations with shared randomness lying within the space spanned by the ... More

Entropy, Convex Optimization, and Competitive Quantum InteractionsNov 12 2005Oct 30 2006This paper has been withdrawn by the author due to errors.

Modeling of heterojunction photovoltaic cells based on ZnO nanowires array and earth-abundant cuprous oxide absorbersDec 09 2010Jan 27 2011As a potential solution for low-cost efficient solar cells, radial junctions consisting of ZnO nanowires arrays embedded in Cu2O thin films have been theoretically modeled. Calculations have been performed to explore the geometric dependence of performance ... More

Contractible open manifolds which embed in no compact, locally connected and locally 1-connected metric spaceSep 07 2018This paper pays a visit to a famous contractible open 3-manifold $W^3$ proposed by R. H. Bing in 1950's. By the finiteness theorem \cite{Hak68}, Haken proved that $W^3$ can embed in no compact 3-manifold. However, until now, the question about whether ... More

The Topological Period-Index Problem over 8-Complexes, ISep 04 2017Jul 04 2019We study the Postnikov tower of the classifying space of a compact Lie group P(n,mn), which gives obstructions to lifting a topological Brauer class of period $n$ to a PU_{mn}-torsor, where the base space is a CW complex of dimension 8. Combined with ... More

Non-Fermi Liquid Behavior in 3+1 Dimensions with PT Invariant Gauge Field: An Renormalization Group ApproachNov 29 1994We introduce a Hamiltonian coupled between a normal Fermi surface and a polarized Maxwell type gauge field.We adopt a {\it calibrated scaling } approach in order to be consistent with the results obtained at $2+1$ dimensions as well as the Weinberg's ... More

An efficient algorithm for solving elliptic problems on percolation clustersJul 31 2019We present an efficient algorithm to solve elliptic Dirichlet problems defined on the cluster of $\mathbb{Z}^d$ supercritical Bernoulli percolation, as a generalization of the iterative method proposed by S. Armstrong, A. Hannukainen, T. Kuusi and J.-C. ... More

On a measure of distance for quantum strategiesAug 27 2010Mar 14 2012The present paper studies an operator norm that captures the distinguishability of quantum strategies in the same sense that the trace norm captures the distinguishability of quantum states or the diamond norm captures the distinguishability of quantum ... More

Graphs with Non-unique Decomposition and Their Associated SurfacesDec 05 2011The ideal (tagged resp.) triangulation of bounded surface with marked points are associated with skew-symmetric (skew-symmetrizable) exchange matrices. An algo- rithm is established to decompose the graph associated to such matrix. There are finite many ... More

Characterization of pseudo-collarable manifolds with boundaryJul 30 2018In this paper we obtain a complete characterization of pseudo-collarable $n$-manifolds for $n\geq 6$. This extends earlier work by Guilbault and Tinsley to allow for manifolds with noncompact boundary. In the same way that their work can be viewed as ... More

Improved Parallel Cache-Oblivious Algorithms for Dynamic Programming and Linear AlgebraSep 25 2018For many cache-oblivious algorithms for dynamic programming and linear algebra, we observe that the key factor that affects the cache complexity is the number of input entries involved in each basic computation cell. In this paper, we propose a level ... More

Some Torsion Classes in the Chow ring and Cohomology of $BPGL_n$Jan 29 2019In the integral cohomology ring of the classifying space of the projective linear group $PGL_n$ (over $\mathbb{C}$), we find a collection of $p$-torsions $y_{p,k}$ of degree $2(p^{k+1}+1)$ for any odd prime divisor $p$ of $n$, and $k\geq 0$. Similarly, ... More

Zero-error communication over adder MACSep 19 2018Adder MAC is a simple noiseless multiple-access channel (MAC), where if users send messages $X_1,\ldots,X_h\in \{0,1\}^n$, then the receiver receives $Y = X_1+\cdots+X_h$ with addition over $\mathbb{Z}$. Communication over the noiseless adder MAC has ... More

A Note on Bounded Biclique Coverings of Complete GraphsNov 09 2015An undirected biclique $K_{a,b}$ is a graph with vertices partitioned into two sets: a set $A$ containing $a$ vertices and a set $B$ containing $b$ vertices such that every vertex in set $A$ is connected to every vertex in set $B$, and such that no two ... More

Integral Finite Surgeries on Knots in $S^3$Jan 27 2014Using the correction terms in Heegaard Floer homology, we prove that if a knot in $S^3$ admits a positive integral $\mathbf{T}$-, $\mathbf{O}$- or $\mathbf{I}$-type surgery, it must have the same knot Floer homology as one of the knots given in our complete ... More

Fluctuation induced non-canonical BCS states: a mechanism for pseudogapDec 31 2010Jan 20 2011We pose the question of what effect the statistical fluctuation causes if it induces a non-unitary evolution. We apply this idea to the BCS model and study fluctuation around the mean-field average. We find that, dynamics of the thermalization influences ... More

A Strong Splitting of the Frobenius Morphism on the Algebra of Distributions of $SL_2$Nov 22 2016Nov 23 2016Let $p$ be a prime number, and let $Dist(SL_2)$ be the algebra of distributions, supported at $1$, on the algebraic group $SL_2$ over $\mathbb{F}_p$. The Frobenius map $Fr:SL_2\to SL_2$ induces a map $Fr:Dist(SL_2)\to Dist(SL_2)$ which is in particular ... More

A note on dimensions of polynomial size circuitsApr 22 2004Apr 22 2004In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every $i\geq 0$, $\Ppoly$ has $i$th order scaled $\pthree$-strong dimension 0. We also show that $\Ppoly^\io$ has $\pthree$-dimension 1/2, ... More

The Topological Period-Index Problem over 8-Complexes, IIMar 14 2018Jun 04 2019We complete the study of the topological period-index problem over 8 dimensional finite CW complexes started in a preceding paper. More precisely, we determine the sharp upper bound of the index of a topological Brauer class $\alpha\in H^3(X;\mathbb{Z})$, ... More

Neyman's C(α) Test for Unobserved HeterogeneityFeb 01 2013Oct 06 2014A unified framework is proposed for tests of unobserved heterogeneity in parametric statistic models based on Neyman's $C(\alpha)$ approach. Such tests are irregular in the sense that the first order derivative of the log likelihood with respect to the ... More

Spanning rigid subgraph packing and sparse subgraph coveringMay 01 2014Jun 12 2018Rigidity, arising in discrete geometry, is the property of a structure that does not flex. Laman provides a combinatorial characterization of rigid graphs in the Euclidean plane, and thus rigid graphs in the Euclidean plane have applications in graph ... More

Quantum Strategies and Local OperationsFeb 26 2010Mar 14 2012This thesis is divided into two parts. In Part I we introduce a new formalism for quantum strategies, which specify the actions of one party in any multi-party interaction involving the exchange of multiple quantum messages among the parties. This formalism ... More

Short Quantum GamesNov 03 2005In this thesis we introduce quantum refereed games, which are quantum interactive proof systems with two competing provers. We focus on a restriction of this model that we call "short quantum games" and we prove an upper bound and a lower bound on the ... More

A Strong Splitting of the Algebra of Distributions of $SL_2$Nov 22 2016Let $p$ be a prime number, and let $Dist(SL_2)$ be the algebra of distributions, supported at $1$, on the algebraic group $SL_2$ over $\mathbb{F}_p$. The Frobenius map $Fr:SL_2\to SL_2$ induces a map $Fr:Dist(SL_2)\to Dist(SL_2)$ which is in particular ... More

Current Status of VHE AstronomyJun 14 1999Very-high-energy astronomy studies the Universe at energies between 30 GeV and 100 TeV. The past decade has seen enormous progress in this field. There are now at least seven known sources of VHE photons. By studying these objects in the VHE regime one ... More

Algebraic integrability of the classical XXZ spin chain with reflecting boundary conditionsAug 22 2014In this paper we analyze the classical XXZ spin chain with reflecting boundaries. We exhibit a system of log-canonical coordinates on the phase space generalizing Sklyanin's separation of variables for the periodic XXZ chain, and use these coordinates ... More

A note on spanoid rankOct 05 2018Nov 21 2018We construct a spanoid $\mathcal{S}$ on $n$ elements with $\textsf{rank}(\mathcal{S}) \ge n^c \textsf{f-rank}(\mathcal{S})$ where $c = \log_5 3 - \log_5 2.5 \approx 0.113283$. This answers a question of Dvir-Gopi-Wigderson [DGW18].

The Topological Period-Index Problem over 8-ComplexesSep 04 2017We study the Postnikov tower of the classifying space of a compact Lie group P(n,mn), which gives obstructions to lifting a topological Brauer class of period $n$ to a PU_{mn}-torsor, where the base space is a CW complex of dimension 8. Combined with ... More

A remark on descent for Coxeter groupsJul 04 2017Jul 06 2017Let $\Gamma$ be a finite Coxeter group with reflection representation $R$. We show that a $\Gamma$-equivariant quasicoherent sheaf on $R$ descends to the quotient space $R//\Gamma$ if it descends to the quotient space $R//\langle s_i\rangle$ for every ... More

On the Cohomology of the Classifying Spaces of Projective Unitary GroupsDec 01 2016Jan 29 2019Let $\mathbf{B}PU_{n}$ be the classifying space of $PU_n$, the projective unitary group of order $n$, for $n>1$. We use the Serre spectral sequence associated to a fiber sequence $\mathbf{B}U_n\rightarrow\mathbf{B}PU_n\rightarrow K(\mathbb{Z},3)$ to determine ... More

Interactive proofs with competing teams of no-signaling proversDec 03 2010Aug 07 2013This paper studies a generalization of multi-prover interactive proofs in which a verifier interacts with two competing teams of provers: one team attempts to convince the verifier to accept while the other attempts to convince the verifier to reject. ... More

The spectral variability in radio-loud quasarsApr 10 2013The spectral variability of a sample of 44 flat-spectrum radio quasars (FSRQs) and 18 steep-spectrum radio quasars (SSRQs) in the SDSS stripe 82 region is investigated. Twenty-five of 44 FSRQs show a bluer-when-brighter trend (BWB), while only one FSRQ ... More

Subspace Iteration Randomization and Singular Value ProblemsAug 10 2014A classical problem in matrix computations is the efficient and reliable approximation of a given matrix by a matrix of lower rank. The truncated singular value decomposition (SVD) is known to provide the best such approximation for any given fixed rank. ... More

Convergence Rates of Neumann problems for Stokes SystemsDec 27 2015This paper studies the convergence rates in $L^2$ and $H^1$ of Neumann problems for Stokes systems with rapidly oscillating periodic coefficients, without any smoothness assumptions on the coefficients.

High order correctors and two-scale expansions in stochastic homogenizationJan 29 2016Oct 03 2016In this paper, we study high order correctors in stochastic homogenization. We consider elliptic equations in divergence form on $\mathbb{Z}^d$, with the random coefficients constructed from i.i.d. random variables. We prove moment bounds on the high ... More

Graph magnitude homology via algebraic Morse theorySep 19 2018We compute magnitude homology of various graphs using algebraic Morse theory. Specifically, we (1) give an alternative proof that trees are diagonal, (2) identify a new class of diagonal graphs, (3) prove that the icosahedral graph is diagonal, and (4) ... More

Random Attractors of Stochastic Lattice Dynamical Systems Driven by Fractional Brownian Motions and its ErratumAug 26 2014Aug 28 2014This paper is devoted to considering the stochastic lattice dynamical systems (SLDS) driven by fractional Brownian motions with Hurst parameter bigger than $1/2$. Under usual dissipativity conditions these SLDS are shown to generate a random dynamical ... More

On Pluri-canonical Systems of Arithmetic SurfacesSep 01 2014Let $S$ be a Dedekind scheme with perfect residue fields at closed points, let $f: X\rightarrow S$ be a minimal regular arithmetic surface of fibre genus at least $2$ and let $f': X'\rightarrow S$ be the canonical model of $f$. It is well known that $\omega_{X'/S}$ ... More

Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the Rayleigh-Taylor sign conditionDec 06 2017We consider a free boundary problem for the axially symmetric incompressible ideal magnetohydrodynamic equations that describes the motion of the plasma in vacuum. Both the plasma magnetic field and vacuum magnetic field are tangent along the plasma-vacuum ... More

Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity conditionNov 23 2017We consider a free boundary problem for the axially symmetric incompressible ideal magnetohydrodynamic equations that describes the motion of the plasma in vacuum. Both the plasma magnetic field and vacuum magnetic field are tangent along the plasma-vacuum ... More

Convergence Rates in Homogenization of Stokes SystemsAug 18 2015Oct 12 2015This paper studies the convergence rates in $L^2$ and $H^1$ of Dirichelt problems for Stokes systems with rapidly oscillating periodic coefficients, without any regularity assumptions on the coefficients.

On the Shrinkable U.S.C. Decomposition Spaces of SpheresDec 04 2013Feb 10 2014Let $G$ be a u.s.c decomposition of $S^n$, $H_G$ denote the set of nondegenerate elements and $\pi$ be the projection of $S^n$ onto $S^n/G$. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with ($n-1$)-sphere frontiers ... More

Characterization of pseudo-collarable manifolds with boundaryJul 30 2018Apr 19 2019In this paper we obtain a complete characterization of pseudo-collarable $n$-manifolds for $n\geq 6$. This extends earlier work by Guilbault and Tinsley to allow for manifolds with noncompact boundary. In the same way that their work can be viewed as ... More

Convergence Rates of Neumann problems for Stokes SystemsDec 27 2015Sep 27 2017This paper studies the convergence rates in $L^2$ and $H^1$ of Neumann problems for Stokes systems with rapidly oscillating periodic coefficients, without any smoothness assumptions on the coefficients.

Enumeration of paths and cycles and e-coefficients of incomparability graphsSep 04 2007We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in terms of elementary ... More

Integrable systems from the classical reflection equationMay 21 2014Jun 08 2015We construct integrable Hamiltonian systems on $G/K$, where $G$ is a quasitriangular Poisson Lie group and $K$ is a Lie subgroup arising as the fixed point set of a group automorphism $\sigma$ of $G$ satisfying the classical reflection equation. In the ... More

The evolution of dusty torus covering factor in quasarsJul 04 2013Jul 15 2013We have assembled a large sample of 5996 quasars at redshift 2.0=< z <= 2.4 (high-z) or 0.7=< z <= 1.1 (low-z) from SDSS data release nine and seven quasar catalogs. The spectral energy distribution (SED) of quasars were constructed by collecting WISE, ... More

The Decomposition Algorithm of Skew-symmetrizable Exchange MatricesFeb 02 2012Feb 04 2012Some skew-symmetrizable integer exchange matrices are associated to ideal (tagged) triangulations of marked bordered surfaces. These exchange matrices admits unfoldings to skew-symmetric matrices. We develop an combinatorial algorithm that determines ... More

Some Results on Reversible Gate Classes Over Non-Binary AlphabetsJun 02 2016We present a collection of results concerning the structure of reversible gate classes over non-binary alphabets, including (1) a reversible gate class over non-binary alphabets that is not finitely generated (2) an explicit set of generators for the ... More

Generalized equivariant model structures on $\mathbf{Cat}^I$May 25 2016Let $I$ be a small category, $\mathcal{C}$ be the category $\mathbf{Cat}$, $\mathbf{Ac}$ or $\mathbf{Pos}$ of small categories, acyclic categories, or posets, respectively. Let $\mathcal{O}$ be a locally small class of objects in $\mathbf{Set}^I$ such ... More

New Variables For Graviton Scattering AmplitudesMay 01 2012Motivated by the success of Hodges' momentum twistor variables in planar Yang-Mills, in this note we introduce a set of new variables, the S variables, which are tailored for gravity (or more generally for theories without color ordering). The S variables ... More

On the Cohomology of the Classifying Spaces of Projective Unitary GroupsDec 01 2016In this paper we construct a spectral sequence converging to the integral cohomology ring $H^{*}(\mathbf{B}PU_{n}; \mathbb{Z})$ for any $n>1$, where $\mathbf{B}PU_{n}$ is the classifying space of the projective unitary group of order $n$. We use this ... More