arXiv subjects

Jiang Hu

Publications and source records attributed to Jiang Hu.

At least 109 records · Page 6Linked to original sources

arXiv2020

ALIGN: A System for Automating Analog Layout

ALIGN ("Analog Layout, Intelligently Generated from Netlists") is an open-source automatic layout generation flow for analog circuits. ALIGN translates an input SPICE netlist to an output GDSII layout, specific to a given technology, as specified by a set of design rules. The flow first automatically detects hierarchies in the circuit netlist and translates layout synthesis to a problem of hierarchical block assembly. At the lowest level, parameterized cells are generated using an abstraction of the design rules; these blocks are then assembled under geometric and electrical constraints to build the circuit layout. ALIGN has been applied to generate layouts for a diverse set of analog circuit families: low frequency analog blocks, wireline circuits, wireless circuits, and power delivery circuits.

cs.AR↗

arXiv2020

Strong consistency of the AIC, BIC, $C_p$ and KOO methods in high-dimensional multivariate linear regression

Variable selection is essential for improving inference and interpretation in multivariate linear regression. Although a number of alternative regressor selection criteria have been suggested, the most prominent and widely used are the Akaike information criterion (AIC), Bayesian information criterion (BIC), Mallow's $C_p$, and their modifications. However, for high-dimensional data, experience has shown that the performance of these classical criteria is not always satisfactory. In the present article, we begin by presenting the necessary and sufficient conditions (NSC) for the strong consistency of the high-dimensional AIC, BIC, and $C_p$, based on which we can identify some reasons for their poor performance. Specifically, we show that under certain mild high-dimensional conditions, if the BIC is strongly consistent, then the AIC is strongly consistent, but not vice versa. This result contradicts the classical understanding. In addition, we consider some NSC for the strong consistency of the high-dimensional kick-one-out (KOO) methods introduced by Zhao et al. (1986) and Nishii et al. (1988). Furthermore, we propose two general methods based on the KOO methods and prove their strong consistency. The proposed general methods remove the penalties while simultaneously reducing the conditions for the dimensions and sizes of the regressors. A simulation study supports our consistency conclusions and shows that the convergence rates of the two proposed general KOO methods are much faster than those of the original methods.

math.ST↗

arXiv2020

Modified Pillai's trace statistics for two high-dimensional sample covariance matrices

The goal of this study was to test the equality of two covariance matrices by using modified Pillai's trace statistics under a high-dimensional framework, i.e., the dimension and sample sizes go to infinity proportionally. In this paper, we introduce two modified Pillai's trace statistics and obtain their asymptotic distributions under the null hypothesis. The benefits of the proposed statistics include the following: (1) the sample size can be smaller than the dimensions; (2) the limiting distributions of the proposed statistics are universal; and (3) we do not restrict the structure of the population covariance matrices. The theoretical results are established under mild and practical assumptions, and their properties are demonstrated numerically by simulations and a real data analysis.

math.ST↗

arXiv2019

The limits of the sample spiked eigenvalues for a high-dimensional generalized Fisher matrix and its applications

A generalized spiked Fisher matrix is considered in this paper. We establish a criterion for the description of the support of the limiting spectral distribution of high-dimensional generalized Fisher matrix and study the almost sure limits of the sample spiked eigenvalues where the population covariance matrices are arbitrary which successively removed an unrealistic condition posed in the previous works, that is, the covariance matrices are assumed to be diagonal or diagonal block-wise structure. In addition, we also give a consistent estimator of the population spiked eigenvalues. A series of simulations are conducted that support the theoretical results and illustrate the accuracy of our estimators.

math.ST↗

arXiv2019

Community Detection Based on the $L_\infty$ convergence of eigenvectors in DCBM

Spectral clustering is one of the most popular algorithms for community detection in network analysis. Based on this rationale, in this paper we give the convergence rate of eigenvectors for the adjacency matrix in the $l_\infty$ norm, under the stochastic block model (BM) and degree corrected stochastic block model (DCBM), adding some mild and rational conditions. We also extend this result to a more general model, presented based on the DCBM such that the value of random variables in the adjacency matrix is not 0 or 1, but an arbitrary real number. During the process of proving the above conclusion, we obtain the relationship of the eigenvalues in the adjacency matrix and the corresponding `population' matrix, which vary in dimension from the community-wise edge probability matrix. Using that result, we can give an estimate of the number of the communities in a known set of network data. Meanwhile we proved the consistency of the estimator. Furthermore, according to the derivation of proof for the convergence of eigenvectors, we propose a new approach to community detection -- Spectral Clustering based on Difference of Ratios of Eigenvectors (SCDRE). Our simulation experiments demonstrate the superiority of our method in community detection.

math.ST↗

arXiv2019

A Brief Introduction to Manifold Optimization

Manifold optimization is ubiquitous in computational and applied mathematics, statistics, engineering, machine learning, physics, chemistry and etc. One of the main challenges usually is the non-convexity of the manifold constraints. By utilizing the geometry of manifold, a large class of constrained optimization problems can be viewed as unconstrained optimization problems on manifold. From this perspective, intrinsic structures, optimality conditions and numerical algorithms for manifold optimization are investigated. Some recent progress on the theoretical results of manifold optimization are also presented.

math.OC↗

arXiv2018

Structured Quasi-Newton Methods for Optimization with Orthogonality Constraints

In this paper, we study structured quasi-Newton methods for optimization problems with orthogonality constraints. Note that the Riemannian Hessian of the objective function requires both the Euclidean Hessian and the Euclidean gradient. In particular, we are interested in applications that the Euclidean Hessian itself consists of a computational cheap part and a significantly expensive part. Our basic idea is to keep these parts of lower computational costs but substitute those parts of higher computational costs by the limited-memory quasi-Newton update. More specically, the part related to Euclidean gradient and the cheaper parts in the Euclidean Hessian are preserved. The initial quasi-Newton matrix is further constructed from a limited-memory Nyström approximation to the expensive part. Consequently, our subproblems approximate the original objective function in the Euclidean space and preserve the orthogonality constraints without performing the so-called vector transports. When the subproblems are solved to sufficient accuracy, both global and local q-superlinear convergence can be established under mild conditions. Preliminary numerical experiments on the linear eigenvalue problem and the electronic structure calculation show the effectiveness of our method compared with the state-of-art algorithms.

math.OC↗

arXiv2018

Optimal modification of the LRT for the equality of two high-dimensional covariance matrices

This paper considers the optimal modification of the likelihood ratio test (LRT) for the equality of two high-dimensional covariance matrices. The classical LRT is not well defined when the dimensions are larger than or equal to one of the sample sizes. In this paper, an optimally modified test that works well in cases where the dimensions may be larger than the sample sizes is proposed. In addition, the test is established under the weakest conditions on the moments and the dimensions of the samples. We also present weakly consistent estimators of the fourth moments, which are necessary for the proposed test, when they are not equal to 3. From the simulation results and real data analysis, we find that the performances of the proposed statistics are robust against affine transformations.

math.ST↗

arXiv2018

High-dimensional covariance matrices in elliptical distributions with application to spherical test

This paper discusses fluctuations of linear spectral statistics of high-dimensional sample covariance matrices when the underlying population follows an elliptical distribution. Such population often possesses high order correlations among their coordinates, which have great impact on the asymptotic behaviors of linear spectral statistics. Taking such kind of dependency into consideration, we establish a new central limit theorem for the linear spectral statistics in this paper for a class of elliptical populations. This general theoretical result has wide applications and, as an example, it is then applied to test the sphericity of elliptical populations.

math.ST↗

arXiv2017

Adaptive Regularized Newton Method for Riemannian Optimization

Optimization on Riemannian manifolds widely arises in eigenvalue computation, density functional theory, Bose-Einstein condensates, low rank nearest correlation, image registration, and signal processing, etc. We propose an adaptive regularized Newton method which approximates the original objective function by the second-order Taylor expansion in Euclidean space but keeps the Riemannian manifold constraints. The regularization term in the objective function of the subproblem enables us to establish a Cauchy-point like condition as the standard trust-region method for proving global convergence. The subproblem can be solved inexactly either by first-order methods or a modified Riemannian Newton method. In the later case, it can further take advantage of negative curvature directions. Both global convergence and superlinear local convergence are guaranteed under mild conditions. Extensive computational experiments and comparisons with other state-of-the-art methods indicate that the proposed algorithm is very promising.

math.OC↗

arXiv2017

Canonical correlation coefficients of high-dimensional Gaussian vectors: finite rank case

Consider a Gaussian vector $\mathbf{z}=(\mathbf{x}',\mathbf{y}')'$, consisting of two sub-vectors $\mathbf{x}$ and $\mathbf{y}$ with dimensions $p$ and $q$ respectively, where both $p$ and $q$ are proportional to the sample size $n$. Denote by $Σ_{\mathbf{u}\mathbf{v}}$ the population cross-covariance matrix of random vectors $\mathbf{u}$ and $\mathbf{v}$, and denote by $S_{\mathbf{u}\mathbf{v}}$ the sample counterpart. The canonical correlation coefficients between $\mathbf{x}$ and $\mathbf{y}$ are known as the square roots of the nonzero eigenvalues of the canonical correlation matrix $Σ_{\mathbf{x}\mathbf{x}}^{-1}Σ_{\mathbf{x}\mathbf{y}}Σ_{\mathbf{y}\mathbf{y}}^{-1}Σ_{\mathbf{y}\mathbf{x}}$. In this paper, we focus on the case that $Σ_{\mathbf{x}\mathbf{y}}$ is of finite rank $k$, i.e. there are $k$ nonzero canonical correlation coefficients, whose squares are denoted by $r_1\geq\cdots\geq r_k>0$. We study the sample counterparts of $r_i,i=1,\ldots,k$, i.e. the largest $k$ eigenvalues of the sample canonical correlation matrix $§_{\mathbf{x}\mathbf{x}}^{-1}§_{\mathbf{x}\mathbf{y}}§_{\mathbf{y}\mathbf{y}}^{-1}§_{\mathbf{y}\mathbf{x}}$, denoted by $λ_1\geq\cdots\geq λ_k$. We show that there exists a threshold $r_c\in(0,1)$, such that for each $i\in\{1,\ldots,k\}$, when $r_i\leq r_c$, $λ_i$ converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by $d_{+}$. When $r_i>r_c$, $λ_i$ possesses an almost sure limit in $(d_{+},1]$. We also obtain the limiting distribution of $λ_i$'s under appropriate normalization. Specifically, $λ_i$ possesses Gaussian type fluctuation if $r_i>r_c$, and follows Tracy-Widom distribution if $r_i<r_c$. Some applications of our results are also discussed.

math.ST↗

arXiv2016

A review of 20 years of naive tests of significance for high-dimensional mean vectors and covariance matrices

In this paper, we will introduce the so called naive tests and give a brief review on the newly development. Naive testing methods are easy to understand and performs robust especially when the dimension is large. In this paper, we mainly focus on reviewing some naive testing methods for the mean vectors and covariance matrices of high dimensional populations and believe this naive test idea can be wildly used in many other testing problems.

math.ST↗

arXiv2015

Convergence of the empirical spectral distribution function of Beta matrices

Let $\mathbf{B}_n=\mathbf {S}_n(\mathbf {S}_n+α_n\mathbf {T}_N)^{-1}$, where $\mathbf {S}_n$ and $\mathbf {T}_N$ are two independent sample covariance matrices with dimension $p$ and sample sizes $n$ and $N$, respectively. This is the so-called Beta matrix. In this paper, we focus on the limiting spectral distribution function and the central limit theorem of linear spectral statistics of $\mathbf {B}_n$. Especially, we do not require $\mathbf {S}_n$ or $\mathbf {T}_N$ to be invertible. Namely, we can deal with the case where $p>\max\{n,N\}$ and $p<n+N$. Therefore, our results cover many important applications which cannot be simply deduced from the corresponding results for multivariate $F$ matrices.

math.PR↗

arXiv2015

On testing the equality of high dimensional mean vectors with unequal covariance matrices

In this article, we focus on the problem of testing the equality of several high dimensional mean vectors with unequal covariance matrices. This is one of the most important problem in multivariate statistical analysis and there have been various tests proposed in the literature. Motivated by \citet{BaiS96E} and \cite{ChenQ10T}, a test statistic is introduced and the asymptomatic distributions under the null hypothesis as well as the alternative hypothesis are given. In addition, it is compared with a test statistic recently proposed by \cite{SrivastavaK13Ta}. It is shown that our test statistic performs much better especially in the large dimensional case.

math.ST↗

arXiv2015

On the semicircular law of large dimensional random quaternion matrices

It is well known that Gaussian symplectic ensemble (GSE) is defined on the space of $n\times n$ quaternion self-dual Hermitian matrices with Gaussian random elements. There is a huge body of literature regarding this kind of matrices. As a natural idea we want to get more universal results by removing the Gaussian condition. For the first step, in this paper we prove that the empirical spectral distribution of the common quaternion self-dual Hermitian matrices tends to semicircular law. The main tool to establish the universal result is given as a lemma in this paper as well.

math.PR↗

arXiv2014

Test of Independence for High-dimensional Random Vectors Based on Block Correlation Matrices

In this paper, we are concerned with the independence test for $k$ high-dimensional sub-vectors of a normal vector, with fixed positive integer $k$. A natural high-dimensional extension of the classical sample correlation matrix, namely block correlation matrix, is raised for this purpose. We then construct the so-called Schott type statistic as our test statistic, which turns out to be a particular linear spectral statistic of the block correlation matrix. Interestingly, the limiting behavior of the Schott type statistic can be figured out with the aid of the Free Probability Theory and the Random Matrix Theory. Specifically, we will bring the so-called real second order freeness for Haar distributed orthogonal matrices, derived in \cite{MP2013}, into the framework of this high-dimensional testing problem. Our test does not require the sample size to be larger than the total or any partial sum of the dimensions of the $k$ sub-vectors. Simulated results show the effect of the Schott type statistic, in contrast to those of the statistics proposed in \cite{JY2013} and \cite{JBZ2013}, is satisfactory. Real data analysis is also used to illustrate our method.

math.ST↗

arXiv2014

Canonical correlation coefficients of high-dimensional normal vectors: finite rank case

Consider a normal vector $\mathbf{z}=(\mathbf{x}',\mathbf{y}')'$, consisting of two sub-vectors $\mathbf{x}$ and $\mathbf{y}$ with dimensions $p$ and $q$ respectively. With $n$ independent observations of $\mathbf{z}$ at hand, we study the correlation between $\mathbf{x}$ and $\mathbf{y}$, from the perspective of the Canonical Correlation Analysis, under the high-dimensional setting: both $p$ and $q$ are proportional to the sample size $n$. In this paper, we focus on the case that $Σ_{\mathbf{x}\mathbf{y}}$ is of finite rank $k$, i.e. there are $k$ nonzero canonical correlation coefficients, whose squares are denoted by $r_1\geq\cdots\geq r_k>0$. Under the additional assumptions $(p+q)/n\to y\in (0,1)$ and $p/q\not\to 1$, we study the sample counterparts of $r_i,i=1,\ldots,k$, i.e. the largest k eigenvalues of the sample canonical correlation matrix $S_{\mathbf{x}\mathbf{x}}^{-1}S_{\mathbf{x}\mathbf{y}}S_{\mathbf{y}\mathbf{y}}^{-1}S_{\mathbf{y}\mathbf{x}}$, namely $λ_1\geq\cdots\geq λ_k$. We show that there exists a threshold $r_c\in(0,1)$, such that for each $i\in\{1,\ldots,k\}$, when $r_i\leq r_c$, $λ_i$ converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by $d_r$. When $r_i>r_c$, $λ_i$ possesses an almost sure limit in $(d_r,1]$, from which we can recover $r_i$ in turn, thus provide an estimate of the latter in the high-dimensional scenario.

math.ST↗

arXiv2014

On the limit of extreme eigenvalues of large dimensional random quaternion matrices

Since E.P.Wigner (1958) established his famous semicircle law, lots of attention has been paid by physicists, probabilists and statisticians to study the asymptotic properties of the largest eigenvalues for random matrices. Bai and Yin (1988) obtained the necessary and sufficient conditions for the strong convergence of the extreme eigenvalues of a Wigner matrix. In this paper, we consider the case of quaternion self-dual Hermitian matrices. We prove the necessary and sufficient conditions for the strong convergence of extreme eigenvalues of quaternion self-dual Hermitian matrices corresponding to the Wigner case.

math.PR↗