Let $\mathbf{W}\in\mathbb{C}^{n\times n}$ be a {\it single-spiked} Wishart matrix in the class $\mathbf{W}\sim \mathcal{CW}_n(m,\mathbf{I}_n+ \theta \mathbf{v}\mathbf{v}^\dagger) $ with $m\geq n$, where $\mathbf{I}_n$ is the $n\times n$ identity matrix, $\mathbf{v}\in\mathbb{C}^{n\times 1}$ is an arbitrary vector with unit Euclidean norm, $\theta\geq 0$ is a non-random parameter, and $(\cdot)^\dagger$ represents the conjugate-transpose operator. Let $\mathbf{u}_1$ and $\mathbf{u}_n$ denote the eigenvectors corresponding to the samllest and the largest eigenvalues of $\mathbf{W}$, respectively. This paper investigates the probability density function (p.d.f.) of the random quantity $Z_{\ell}^{(n)}=\left|\mathbf{v}^\dagger\mathbf{u}_\ell\right|^2\in(0,1)$ for $\ell=1,n$. In particular, we derive a finite dimensional closed-form p.d.f. for $Z_{1}^{(n)}$ which is amenable to asymptotic analysis as $m,n$ diverges with $m-n$ fixed. It turns out that, in this asymptotic regime, the scaled random variable $nZ_{1}^{(n)}$ converges in distribution to $\chi^2_2/2(1+\theta)$, where $\chi_2^2$ denotes a chi-squared random variable with two degrees of freedom. This reveals that $\mathbf{u}_1$ can be used to infer information about the spike. On the other hand, the finite dimensional p.d.f. of $Z_{n}^{(n)}$ is expressed as a double integral in which the integrand contains a determinant of a square matrix of dimension $(n-2)$. Although a simple solution to this double integral seems intractable, for special configurations of $n=2,3$, and $4$, we obtain closed-form expressions.
相關內容
Integration：Integration, the VLSI Journal。
Explanation：集成，VLSI雜志。
Publisher：Elsevier。
SIT：
We deal with a long-standing problem about how to design an energy-stable numerical scheme for solving the motion of a closed curve under {\sl anisotropic surface diffusion} with a general anisotropic surface energy $\gamma(\boldsymbol{n})$ in two dimensions, where $\boldsymbol{n}$ is the outward unit normal vector. By introducing a novel symmetric positive definite surface energy matrix $Z_k(\boldsymbol{n})$ depending on the Cahn-Hoffman $\boldsymbol{\xi}$-vector and a stabilizing function $k(\boldsymbol{n})$, we first reformulate the anisotropic surface diffusion into a conservative form and then derive a new symmetrized variational formulation for the anisotropic surface diffusion with weakly or strongly anisotropic surface energies. A semi-discretization in space for the symmetrized variational formulation is proposed and its area (or mass) conservation and energy dissipation are proved. The semi-discretization is then discretized in time by either an implicit structural-preserving scheme (SP-PFEM) which preserves the area in the discretized level or a semi-implicit energy-stable method (ES-PFEM) which needs only solve a linear system at each time step. Under a relatively simple and mild condition on $\gamma(\boldsymbol{n})$, we show that both SP-PFEM and ES-PFEM are unconditionally energy-stable for almost all anisotropic surface energies $\gamma(\boldsymbol{n})$ arising in practical applications. Specifically, for several commonly-used anisotropic surface energies, we construct $Z_k(\boldsymbol{n})$ explicitly. Finally, extensive numerical results are reported to demonstrate the high performance of the proposed numerical schemes.
The eigenvalues of Toeplitz matrices $T_{n}(f)$ with a real-valued symbol $f$, satisfying some conditions and tracing out a simple loop over the interval $[-\pi,\pi]$, are known to admit an asymptotic expansion with the form \[ \lambda_{j}(T_{n}(f))=f(d_{j,n})+c_{1}(d_{j,n})h+c_{2}(d_{j,n})h^{2}+O(h^{3}), \] where $h=\frac{1}{n+1}$, $d_{j,n}=\pi j h$, and $c_k$ are some bounded coefficients depending only on $f$. The numerical results presented in the literature suggests that the effective conditions for the expansion to hold are weaker and reduce to an even character of $f$, to a fixed smoothness, and to its monotonicity over $[0,\pi]$. \\ In this note we investigate the superposition caused over this expansion, when considering a linear combination of symbols that is \[ \lambda_{j}\big(T_{n}(f_0)+\beta_{n}^{(1)} T_{n}(f_{1}) + \beta_{n}^{(2)} T_{n}(f_{2}) +\cdots\big), \] where $ \beta_{n}^{(t)}=o\big(\beta_{n}^{(s)}\big)$ if $t>s$ and the symbols $f_{j}$ are either simple loop or satisfy the weaker conditions mentioned before. We prove that the asymptotic expansion holds also in this setting under mild assumptions and we show numerically that there is much more to investigate, opening the door to linear in time algorithms for the computation of eigenvalues of large matrices of this type. The problem is of concrete interest in particular in the case where the coefficients of the linear combination are functions of $h$, considering spectral features of matrices stemming from the numerical approximation of standard differential operators and distributed order fractional differential equations, via local methods such as Finite Differences, Finite Elements, Isogeometric Analysis etc.
The problem of finding the unique low dimensional decomposition of a given matrix has been a fundamental and recurrent problem in many areas. In this paper, we study the problem of seeking a unique decomposition of a low rank matrix $Y\in \mathbb{R}^{p\times n}$ that admits a sparse representation. Specifically, we consider $Y = A X\in \mathbb{R}^{p\times n}$ where the matrix $A\in \mathbb{R}^{p\times r}$ has full column rank, with $r < \min\{n,p\}$, and the matrix $X\in \mathbb{R}^{r\times n}$ is element-wise sparse. We prove that this sparse decomposition of $Y$ can be uniquely identified, up to some intrinsic signed permutation. Our approach relies on solving a nonconvex optimization problem constrained over the unit sphere. Our geometric analysis for the nonconvex optimization landscape shows that any {\em strict} local solution is close to the ground truth solution, and can be recovered by a simple data-driven initialization followed with any second order descent algorithm. At last, we corroborate these theoretical results with numerical experiments.
This paper proposes a regularization of the Monge-Amp\`ere equation in planar convex domains through uniformly elliptic Hamilton-Jacobi-Bellman equations. The regularized problem possesses a unique strong solution $u_\varepsilon$ and is accessible to the discretization with finite elements. This work establishes locally uniform convergence of $u_\varepsilon$ to the convex Alexandrov solution $u$ to the Monge-Amp\`ere equation as the regularization parameter $\varepsilon$ approaches $0$. A mixed finite element method for the approximation of $u_\varepsilon$ is proposed, and the regularized finite element scheme is shown to be locally uniformly convergent. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions $u$.
Eigenvector centrality is one of the outstanding measures of central tendency in graph theory. In this paper we consider the problem of calculating eigenvector centrality of graph partitioned into components and how this partitioning can be used. Two cases are considered; first where the a single component in the graph has the dominant eigenvalue, secondly when there are at least two components that share the dominant eigenvalue for the graph. In the first case we implement and compare the method to the usual approach (power method) for calculating eigenvector centrality while in the second case with shared dominant eigenvalues we show some theoretical and numerical results. Keywords: Eigenvector centrality, power iteration, graph, strongly connected component.
Resolution of 2D reconstruction of functions with nonsmooth edges from discrete Radon transform data
Let $f$ be an unknown function in $\mathbb R^2$, and $f_\epsilon$ be its reconstruction from discrete Radon transform data, where $\epsilon$ is the data sampling rate. We study the resolution of reconstruction when $f$ has a jump discontinuity along a nonsmooth curve $\mathcal S_\epsilon$. The assumptions are that (a) $\mathcal S_\epsilon$ is an $O(\epsilon)$-size perturbation of a smooth curve $\mathcal S$, and (b) $\mathcal S_\epsilon$ is Holder continuous with some exponent $\gamma\in(0,1]$. We compute the Discrete Transition Behavior (or, DTB) defined as the limit $\text{DTB}(\check x):=\lim_{\epsilon\to0}f_\epsilon(x_0+\epsilon\check x)$, where $x_0$ is generic. We illustrate the DTB by two sets of numerical experiments. In the first set, the perturbation is a smooth, rapidly oscillating sinusoid, and in the second - a fractal curve. The experiments reveal that the match between the DTB and reconstruction is worse as $\mathcal S_\epsilon$ gets more rough. This is in agreement with the proof of the DTB, which suggests that the rate of convergence to the limit is $O(\epsilon^{\gamma/2})$. We then propose a new DTB, which exhibits an excellent agreement with reconstructions. Investigation of this phenomenon requires computing the rate of convergence for the new DTB. This, in turn, requires completely new approaches. We obtain a partial result along these lines and formulate a conjecture that the rate of convergence of the new DTB is $O(\epsilon^{1/2}\ln(1/\epsilon))$.
The noncentral Wishart distribution has become more mainstream in statistics as the prevalence of applications involving sample covariances with underlying multivariate Gaussian populations as dramatically increased since the advent of computers. Multiple sources in the literature deal with local approximations of the noncentral Wishart distribution with respect to its central counterpart. However, no source has yet developed explicit local approximations for the (central) Wishart distribution in terms of a normal analogue, which is important since Gaussian distributions are at the heart of the asymptotic theory for many statistical methods. In this paper, we prove a precise asymptotic expansion for the ratio of the Wishart density to the symmetric matrix-variate normal density with the same mean and covariances. The result is then used to derive an upper bound on the total variation between the corresponding probability measures and to find the pointwise variance of a new density estimator on the space of positive definite matrices with a Wishart asymmetric kernel. For the sake of completeness, we also find expressions for the pointwise bias of our new estimator, the pointwise variance as we move towards the boundary of its support, the mean squared error, the mean integrated squared error away from the boundary, and we prove its asymptotic normality.
Empirical likelihood enables a nonparametric, likelihood-driven style of inference without restrictive assumptions routinely made in parametric models. We develop a framework for applying empirical likelihood to the analysis of experimental designs, addressing issues that arise from blocking and multiple hypothesis testing. In addition to popular designs such as balanced incomplete block designs, our approach allows for highly unbalanced, incomplete block designs. Based on all these designs, we derive an asymptotic multivariate chi-square distribution for a set of empirical likelihood test statistics. Further, we propose two single-step multiple testing procedures: asymptotic Monte Carlo and nonparametric bootstrap. Both procedures asymptotically control the generalized family-wise error rate and efficiently construct simultaneous confidence intervals for comparisons of interest without explicitly considering the underlying covariance structure. A simulation study demonstrates that the performance of the procedures is robust to violations of standard assumptions of linear mixed models. Significantly, considering the asymptotic nature of empirical likelihood, the nonparametric bootstrap procedure performs well even for small sample sizes. We also present an application to experiments on a pesticide. Supplementary materials for this article are available online.
We design a new algorithm for solving parametric systems having finitely many complex solutions for generic values of the parameters. More precisely, let $f = (f_1, \ldots, f_m)\subset \mathbb{Q}[y][x]$ with $y = (y_1, \ldots, y_t)$ and $x = (x_1, \ldots, x_n)$, $V\subset \mathbb{C}^{t+n}$ be the algebraic set defined by $f$ and $\pi$ be the projection $(y, x) \to y$. Under the assumptions that $f$ admits finitely many complex roots for generic values of $y$ and that the ideal generated by $f$ is radical, we solve the following problem. On input $f$, we compute semi-algebraic formulas defining semi-algebraic subsets $S_1, \ldots, S_l$ of the $y$-space such that $\cup_{i=1}^l S_i$ is dense in $\mathbb{R}^t$ and the number of real points in $V\cap \pi^{-1}(\eta)$ is invariant when $\eta$ varies over each $S_i$. This algorithm exploits properties of some well chosen monomial bases in the algebra $\mathbb{Q}(y)[x]/I$ where $I$ is the ideal generated by $f$ in $\mathbb{Q}(y)[x]$ and the specialization property of the so-called Hermite matrices. This allows us to obtain compact representations of the sets $S_i$ by means of semi-algebraic formulas encoding the signature of a symmetric matrix. When $f$ satisfies extra genericity assumptions, we derive complexity bounds on the number of arithmetic operations in $\mathbb{Q}$ and the degree of the output polynomials. Let $d$ be the maximal degree of the $f_i$'s and $D = n(d-1)d^n$, we prove that, on a generic $f=(f_1,\ldots,f_n)$, one can compute those semi-algebraic formulas with $O^~( \binom{t+D}{t}2^{3t}n^{2t+1} d^{3nt+2(n+t)+1})$ operations in $\mathbb{Q}$ and that the polynomials involved have degree bounded by $D$. We report on practical experiments which illustrate the efficiency of our algorithm on generic systems and systems from applications. It allows us to solve problems which are out of reach of the state-of-the-art.
In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.
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