arXiv subjects

Lun Zhang

Publications and source records attributed to Lun Zhang.

At least 55 records · Page 3Linked to original sources

arXiv2018

Auto-Context R-CNN

Region-based convolutional neural networks (R-CNN)~\cite{fast_rcnn,faster_rcnn,mask_rcnn} have largely dominated object detection. Operators defined on RoIs (Region of Interests) play an important role in R-CNNs such as RoIPooling~\cite{fast_rcnn} and RoIAlign~\cite{mask_rcnn}. They all only utilize information inside RoIs for RoI prediction, even with their recent deformable extensions~\cite{deformable_cnn}. Although surrounding context is well-known for its importance in object detection, it has yet been integrated in R-CNNs in a flexible and effective way. Inspired by the auto-context work~\cite{auto_context} and the multi-class object layout work~\cite{nms_context}, this paper presents a generic context-mining RoI operator (i.e., \textit{RoICtxMining}) seamlessly integrated in R-CNNs, and the resulting object detection system is termed \textbf{Auto-Context R-CNN} which is trained end-to-end. The proposed RoICtxMining operator is a simple yet effective two-layer extension of the RoIPooling or RoIAlign operator. Centered at an object-RoI, it creates a $3\times 3$ layout to mine contextual information adaptively in the $8$ surrounding context regions on-the-fly. Within each of the $8$ context regions, a context-RoI is mined in term of discriminative power and its RoIPooling / RoIAlign features are concatenated with the object-RoI for final prediction. \textit{The proposed Auto-Context R-CNN is robust to occlusion and small objects, and shows promising vulnerability for adversarial attacks without being adversarially-trained.} In experiments, it is evaluated using RoIPooling as the backbone and shows competitive results on Pascal VOC, Microsoft COCO, and KITTI datasets (including $6.9\%$ mAP improvements over the R-FCN~\cite{rfcn} method on COCO \textit{test-dev} dataset and the first place on both KITTI pedestrian and cyclist detection as of this submission).

cs.CV↗

arXiv2018

Gap probability at the hard edge for random matrix ensembles with pole singularities in the potential

We study the Fredholm determinant of an integrable operator acting on the interval $(0,s)$ whose kernel is constructed out of a hierarchy of higher order analogues to the Painlevé III equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probability at the hard edge of unitary invariant random matrix ensembles perturbed by poles of order $k$ in the double scaling regime. Using the Riemann-Hilbert method, we obtain the large $s$ asymptotics of the Fredholm determinant. Moreover, we derive a Painlevé type formula of the Fredholm determinant, which is expressed in terms of an explicit integral involving a solution to the coupled Painlevé III system.

math-ph↗

arXiv2017

Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and particular Hermitian anti-self dual matrices, involving both sums and products. As a consequence of these results, the eigenvalue probability density function of the random product structure $X_M \cdots X_1( iA) X_1^T \cdots X_M^T$, where each $X_i$ is a standard real Gaussian matrix, and $A$ is a real anti-symmetric matrix can be determined. For $M=1$ and $A$ the bidiagonal anti-symmetric matrix with 1's above the diagonal, this reclaims results of Defosseux. For general $M$, and this choice of $A$, or $A$ itself a standard Gaussian anti-symmetric matrix, the eigenvalue distribution is shown to coincide with that of the squared singular values for the product of certain complex Gaussian matrices first studied by Akemann et al. As a point of independent interest, we also include a self-contained diffusion equation derivation of the orthogonal and symplectic Harish-Chandra integrals.

math-ph↗

arXiv2017

Object Detection via Aspect Ratio and Context Aware Region-based Convolutional Networks

Jointly integrating aspect ratio and context has been extensively studied and shown performance improvement in traditional object detection systems such as the DPMs. It, however, has been largely ignored in deep neural network based detection systems. This paper presents a method of integrating a mixture of object models and region-based convolutional networks for accurate object detection. Each mixture component accounts for both object aspect ratio and multi-scale contextual information explicitly: (i) it exploits a mixture of tiling configurations in the RoI pooling to remedy the warping artifacts caused by a single type RoI pooling (e.g., with equally-sized 7 x 7 cells), and to respect the underlying object shapes more; (ii) it "looks from both the inside and the outside of a RoI" by incorporating contextual information at two scales: global context pooled from the whole image and local context pooled from the surrounding of a RoI. To facilitate accurate detection, this paper proposes a multi-stage detection scheme for integrating the mixture of object models, which utilizes the detection results of the model at the previous stage as the proposals for the current in both training and testing. The proposed method is called the aspect ratio and context aware region-based convolutional network (ARC-R-CNN). In experiments, ARC-R-CNN shows very competitive results with Faster R-CNN [41] and R-FCN [10] on two datasets: the PASCAL VOC and the Microsoft COCO. It obtains significantly better mAP performance using high IoU thresholds on both datasets.

cs.CV↗

arXiv2017

Jacobi polynomials on the Bernstein ellipse

In this paper, we are concerned with Jacobi polynomials $P_n^{(α,β)}(x)$ on the Bernstein ellipse with motivation mainly coming from recent studies of convergence rate of spectral interpolation. An explicit representation of $P_n^{(α,β)}(x)$ is derived in the variable of parametrization. This formula further allows us to show that the maximum value of $\left|P_n^{(α,β)}(z)\right|$ over the Bernstein ellipse is attained at one of the endpoints of the major axis if $α+β\geq -1$. For the minimum value, we are able to show that for a large class of Gegenbauer polynomials (i.e., $α=β$), it is attained at two endpoints of the minor axis. These results particularly extend those previously known only for some special cases. Moreover, we obtain a more refined asymptotic estimate for Jacobi polynomials on the Bernstein ellipse.

math.NA↗

arXiv2016

Mixed type multiple orthogonal polynomials associated with the modified Bessel functions and products of two coupled random matrices

We consider mixed type multiple orthogonal polynomials associated with a system of weight functions consisting of two vectors. One vector is defined in terms of scaled modified Bessel function of the first kind $I_μ$ and $I_{μ+1}$, the other vector is defined in terms of scaled modified Bessel function of the second kind $K_ν$ and $K_{ν+1}$. We show that the corresponding mixed type multiple orthogonal polynomials exist. For the special case that each multi-index is on or close to the diagonal, basic properties of the polynomials and their linear forms are investigated, which include explicit formulas, integral representations, differential properties, limiting forms and recurrence relations. It comes out that, for specified parameters, the linear forms of these mixed type multiple orthogonal polynomials can be interpreted as biorthogonal functions encountering in recent studies of products of two coupled random matrices. This particularly implies a Riemann-Hilbert characterization of the correlation kernel, which provides an alternative way for further asymptotic analysis.

math.CA↗

arXiv2016

On Wright's generalized Bessel kernel

In this paper, we consider the Wright's generalized Bessel kernel $K^{(α,θ)}(x,y)$ defined by $$θx^α\int_0^1J_{\frac{α+1}θ,\frac{1}θ}(ux)J_{α+1,θ}((uy)^θ)u^α\,\mathrm{d} u, \qquad α>-1, \qquad θ>0,$$ where $$J_{a,b}(x)=\sum_{j=0}^\infty\frac{(-x)^j}{j!Γ(a+bj)},\qquad a\in\mathbb{C},\qquad b>-1,$$ is Wright's generalization of the Bessel function. This non-symmetric kernel, which generalizes the classical Bessel kernel (corresponding to $θ=1$) in random matrix theory, is the hard edge scaling limit of the correlation kernel for certain Muttalib-Borodin ensembles. We show that, if $θ$ is rational, i.e., $θ=\frac{m}{n}$ with $m,n\in\mathbb{N}$, $gcd(m,n)=1$, and $α> m-1-\frac{m}{n}$, the Wright's generalized Bessel kernel is integrable in the sense of Its-Izergin-Korepin-Slavnov. We then come to the Fredholm determinant of this kernel over the union of several scaled intervals, which can also be interpreted as the gap probability (the probability of finding no particles) on these intervals. The integrable structure allows us to obtain a system of coupled partial differential equations associated with the corresponding Fredholm determinant as well as a Hamiltonian interpretation. As a consequence, we are able to represent the gap probability over a single interval $(0,s)$ in terms of a solution of a system of nonlinear ordinary differential equations.

math-ph↗

arXiv2016

Biaxial strain tuned thermoelectric properties in monolayer $\mathrm{PtSe_2}$

Strain engineering is a very effective method to tune electronic, optical, topological and thermoelectric properties of materials. In this work, we systematically study biaxial strain dependence of electronic structures and thermoelectric properties (both electron and phonon parts) of monolayer $\mathrm{PtSe_2}$ with generalized gradient approximation (GGA) plus spin-orbit coupling (SOC) for electron part and GGA for phonon part. Calculated results show that compressive or tensile strain can induce conduction band minimum (CBM) or valence band maximum (VBM) transition, which produces important effects on Seebeck coefficient. It is found that compressive or tensile strain can induce significantly enhanced n- or p-type Seebeck coefficient at the critical strain of CBM or VBM transition, which can be explained by strain-induced band convergence. Another essential strain effect is that tensile strain can produce significantly reduced lattice thermal conductivity, and the room temperature lattice thermal conductivity at the strain of -4.02\% can decrease by about 60\% compared to unstrained one, which is very favorable for high $ZT$. To estimate efficiency of thermoelectric conversion, the figure of merit $ZT$ can be obtained by empirical scattering time $τ$. Calculated $ZT$ values show that strain indeed is a very effective strategy to achieve enhanced thermoelectric properties, especially for p-type doping. Tuning thermoelectric properties with strain also can be applied to other semiconducting transition-metal dichalcogenide monolayers $\mathrm{MX_2}$ (M=Zr, Hf, Mo, W and Pt; X=S, Se and Te).

cond-mat.mtrl-sci↗

arXiv2015

Local universality in biorthogonal Laguerre ensembles

We consider $n$ particles $0\leq x_1 -1,~~ θ>0, $$ where $Z_n$ is the normalization constant. This distribution arises in the context of modeling disordered conductors in the metallic regime, and can also be realized as the distribution for squared singular values of certain triangular random matrices. We give a double contour integral formula for the correlation kernel, which allows us to establish universality for the local statistics of the particles, namely, the bulk universality and the soft edge universality via the sine kernel and the Airy kernel, respectively. In particular, our analysis also leads to new double contour integral representations of scaling limits at the origin (hard edge), which are equivalent to those found in the classical work of Borodin. We conclude this paper by relating the correlation kernels to those appearing in recent studies of products of $M$ Ginibre matrices for the special cases $θ=M\in\mathbb{N}$.

math-ph↗

arXiv2015

Location of Poles for the Hastings-McLeod Solution to the Second Painlevé Equation

We show that the well-known Hastings-McLeod solution to the second Painlevé equation is pole-free in the region $\arg x \in [-\fracπ{3},\fracπ{3}]\cup [\frac{2π}{3},\frac{ 4 π}{3}]$, which proves an important special case of a general conjecture concerning pole distributions of Painlevé transcedents proposed by Novokshenov. Our strategy is to construct explicit quasi-solutions approximating the Hastings-McLeod solution in different regions of the complex plane, and estimate the errors rigorously. The main idea is very similar to the one used to prove Dubrovin's conjecture for the first Painlevé equation, but there are various technical improvements.

math.CA↗

arXiv2015

Bulk and soft-edge universality for singular values of products of Ginibre random matrices

It has been shown by Akemann, Ipsen and Kieburg that the squared singular values of products of $M$ rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that admits a representation in terms of Meijer G-functions. We prove the universality of the local statistics of the squared singular values, namely, the bulk universality given by the sine kernel and the edge universality given by the Airy kernel. The proof is based on the asymptotic analysis for the double contour integral representation of the correlation kernel. Our strategy can be generalized to deal with other models of products of random matrices introduced recently and to establish similar universal results. Two more examples are investigated, one is the product of $M$ Ginibre matrices and the inverse of $K$ Ginibre matrices studied by Forrester, and the other one is the product of $M-1$ Ginibre matrices with one truncated unitary matrix considered by Kuijlaars and Stivigny.

math.PR↗

arXiv2014

On Certain Wronskians of Multiple Orthogonal Polynomials

We consider determinants of Wronskian type whose entries are multiple orthogonal polynomials associated with a path connecting two multi-indices. By assuming that the weight functions form an algebraic Chebyshev (AT) system, we show that the polynomials represented by the Wronskians keep a constant sign in some cases, while in some other cases oscillatory behavior appears, which generalizes classical results for orthogonal polynomials due to Karlin and Szegő. There are two applications of our results. The first application arises from the observation that the $m$-th moment of the average characteristic polynomials for multiple orthogonal polynomial ensembles can be expressed as a Wronskian of the type II multiple orthogonal polynomials. Hence, it is straightforward to obtain the distinct behavior of the moments for odd and even $m$ in a special multiple orthogonal ensemble - the AT ensemble. As the second application, we derive some Turán type inequalities for multiple Hermite and multiple Laguerre polynomials (of two kinds). Finally, we study numerically the geometric configuration of zeros for the Wronskians of these multiple orthogonal polynomials. We observe that the zeros have regular configurations in the complex plane, which might be of independent interest.

math.CA↗

arXiv2014

Multiple orthogonal polynomials associated with an exponential cubic weight

We consider multiple orthogonal polynomials associated with the exponential cubic weight e^{-x^3} over two contours in the complex plane. We study the basic properties of these polynomials, including the Rodrigues formula and nearest-neighbor recurrence relations. It turns out that the recurrence coefficients are related to a discrete Painlevé equation. The asymptotics of the recurrence coefficients, the ratio of the diagonal multiple orthogonal polynomials and the (scaled) zeros of these polynomials are also investigated.

math.CA↗

arXiv2013

Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits

Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M=2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy-Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.

math-ph↗

arXiv2013

A note on the limiting mean distribution of singular values for products of two Wishart random matrices

The product of M complex random Gaussian matrices of size N has recently been studied by Akemann, Kieburg and Wei. They showed that, for fixed M and N, the joint probability distribution for the squared singular values of the product matrix forms a determinantal point process with a correlation kernel determined by certain biorthogonal polynomials that can be explicitly constructed. We find that, in the case M=2, the relevant biorthogonal polynomials are actually special cases of multiple orthogonal polynomials associated with Macdonald functions (modified Bessel functions of the second kind) which was first introduced by Van Assche and Yakubovich. With known results on asymptotic zero distribution of these polynomials and general theory on multiple orthogonal polynomial ensembles, it is then easy to obtain an explicit expression for the distribution of squared singular values for the product of two complex random Gaussian matrices in the limit of large matrix dimensions.

math-ph↗

arXiv2013

Universality and critical behavior in the chiral two-matrix model

We study the chiral two-matrix model with polynomial potential functions $V$ and $W$, which was introduced by Akemann, Damgaard, Osborn and Splittorff. We show that the squared singular values of each of the individual matrices in this model form a determinantal point process with correlation kernel determined by a matrix-valued Riemann-Hilbert problem. The size of the Riemann-Hilbert matrix depends on the degree of the potential function $W$ (or $V$ respectively). In this way we obtain the chiral analogue of a result of Kuijlaars-McLaughlin for the non-chiral two-matrix model. The Gaussian case corresponds to $V,W$ being linear. For the case where $W(y)=y^2/2+αy$ is quadratic, we derive the large $n$-asymptotics of the Riemann-Hilbert problem by means of the Deift-Zhou steepest descent method. This proves universality in this case. An important ingredient in the analysis is a third-order differential equation. Finally we show that if also $V(x)=x$ is linear, then a multi-critical limit of the kernel exists which is described by a $4\times 4$ matrix-valued Riemann-Hilbert problem associated to the Painlevé II equation $q"(x) = xq(x)+2q^3(x)-ν-1/2$. In this way we obtain the chiral analogue of a recent result by Duits and the second author.

math-ph↗

arXiv2012

The generalized Krawtchouk polynomials and the fifth Painlevé equation

We study the recurrence coefficients of the orthogonal polynomials with respect to a semi-classical extension of the Krawtchouk weight. We derive a coupled discrete system for these coefficients and show that they satisfy the fifth Painlevé equation when viewed as functions of one of the parameters in the weight.

math.CA↗

arXiv2012

Transitions between critical kernels: from the tacnode kernel and critical kernel in the two-matrix model to the Pearcey kernel

In this paper we study two multicritical correlation kernels and prove that they converge to the Pearcey kernel in a certain double scaling limit. The first kernel appears in a model of non-intersecting Brownian motions at a tacnode. The second arises as a triple scaling limit of the eigenvalue correlation kernel in the Hermitian two-matrix model with quartic/quadratic potentials. The two kernels are different but can be expressed in terms of the same tacnode Riemann-Hilbert problem. The proof is based on a steepest descent analysis of this Riemann-Hilbert problem. A special feature in the analysis is the introduction of an explicit meromorphic function on a Riemann surface with specified sheet structure.

math-ph↗