Times 2n Research Articles

Sparse random hypergraphs: non-backtracking spectra and community detection

Abstract We consider the community detection problem in a sparse $q$-uniform hypergraph $G$, assuming that $G$ is generated according to the Hypergraph Stochastic Block Model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten–Stigum detection threshold conjectured by Angelini et al. (2015, Spectral detection on sparse hypergraphs. In: 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton). IEEE, pp. 66–73). We characterize the spectrum of the non-backtracking operator for the sparse HSBM and provide an efficient dimension reduction procedure using the Ihara–Bass formula for hypergraphs. As a result, community detection for the sparse HSBM on $n$ vertices can be reduced to an eigenvector problem of a $2n\times 2n$ non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. To the best of our knowledge, this is the first provable and efficient spectral algorithm that achieves the conjectured threshold for HSBMs with $r$ blocks generated according to a general symmetric probability tensor.

Block perturbation of symplectic matrices in Williamson’s theorem

AbstractWilliamson’s theorem states that for any $2n \times 2n$ real positive definite matrix A, there exists a $2n \times 2n$ real symplectic matrix S such that $S^TAS=D \oplus D$ , where D is an $n\times n$ diagonal matrix with positive diagonal entries known as the symplectic eigenvalues of A. Let H be any $2n \times 2n$ real symmetric matrix such that the perturbed matrix $A+H$ is also positive definite. In this paper, we show that any symplectic matrix $\tilde {S}$ diagonalizing $A+H$ in Williamson’s theorem is of the form $\tilde {S}=S Q+\mathcal {O}(\|H\|)$ , where Q is a $2n \times 2n$ real symplectic as well as orthogonal matrix. Moreover, Q is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of A. Consequently, we show that $\tilde {S}$ and S can be chosen so that $\|\tilde {S}-S\|=\mathcal {O}(\|H\|)$ . Our results hold even if A has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear Algebra Appl., 525:45–58, 2017].

An Anti-Steganalysis HEVC Video Steganography With High Performance Based on CNN and PU Partition Modes

The steganography research of videos leads to excellent communication methods for transmitting secret message, and high efficiency video coding(HEVC) video is one popular steganographic carrier. This article proposes a prediction unit(PU) based wide residual-net steganography(PWRN) for HEVC videos. The visual quality distortion of modifying PUs is theoretically analyzed, which illustrates that modifying PUs only has a little negative effect on visual quality. Therefore, the data hiding method in this article allows to modify all types of PUs except for <inline-formula><tex-math notation="LaTeX">$2N\times 2N$</tex-math></inline-formula> to each other according to the secret data. In this way, high embedding efficiency is achieved, and the PU distributions in stego-videos can be kept similar to those of cover-videos, which is essential for resisting steganalysis. Meanwhile, a super-resolution convolutional neural network(CNN) with wide residual-net filter(WRNF) is proposed to replace the in-loop filter in HEVC for reconstructing I-pictures, which results in more precisely predicted P-pictures, and it further leads to less bitrate cost and better visual quality of stego-videos. The experimental results show that the proposed PWRN successfully resists the latest PU-targeted steganalysis algorithms, and compared with the state-of-the-art work, PWRN has achieved the lowest bitrate cost and the highest visual quality under the same capacity.

The $L^p$ boundedness of the wave operators for matrix Schrödinger equations

We prove that the wave operators for n \times n matrix Schrödinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces L^p(\mathbb R^+, \mathbb C^n), 1 &lt; p &lt; \infty, for slowly decaying selfadjoint matrix potentials V that satisfy the condition \int_{0}^{\infty}(1+x) |V(x)|\: dx &lt; \infty. Moreover, assuming that \int_{0}^{\infty }(1+x^\gamma) |V(x)|\: dx &lt; \infty, \gamma &gt; \frac{5}{2}, and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in L^1(\mathbb R^+, \mathbb C^n) and in L^\infty(\mathbb R^+, \mathbb C^n). We also prove that the wave operators for n\times n matrix Schrödinger equations on the line are bounded in the spaces L^p(\mathbb R, \mathbb C^n), 1 &lt; p &lt; \infty, assuming that the perturbation consists of a point interaction at the origin and of a potential \mathcal V that satisfies the condition \int_{-\infty}^{\infty}(1+|x|)|\mathcal V(x)|\: dx &lt; \infty. Further, assuming that \int_{-\infty}^{\infty }(1+|x|^\gamma) |\mathcal V(x)|\: dx &lt; \infty, \gamma &gt; \frac{5}{2}, and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in L^1(\mathbb R, \mathbb C^n) and in L^\infty(\mathbb R, \mathbb C^n). We obtain our results for n\times n matrix Schrödinger equations on the line from the results for 2n\times 2n matrix Schrödinger equations on the half line.

On the Local Eigenvalue Statistics for Random Band Matrices in the Localization Regime

We study the local eigenvalue statistics $\xi_{\omega,E}^N$ associated with the eigenvalues of one-dimensional, $(2N+1) \times (2N+1)$ random band matrices with independent, identically distributed, real random variables and band width growing as $N^\alpha$, for $0 < \alpha < \frac{1}{2}$. We consider the limit points associated with the random variables $\xi_{\omega,E}^N [I]$, for $I \subset \mathbb{R}$, and $E \in (-2,2)$. For Gaussian distributed random variables with $0 \leq \alpha < \frac{1}{7}$, we prove that this family of random variables has nontrivial limit points for almost every $E \in (-2,2)$, and that these limit points are Poisson distributed with positive intensities. The proof is based on an analysis of the characteristic functions of the random variables $\xi_{\omega,E}^N [I]$ and associated quantities related to the intensities, as $N$ tends towards infinity, and employs known localization bounds of \cite{schenker, peled, et. al.}, and the strong Wegner and Minami estimates \cite{peled, et. al.}. Our more general result applies to random band matrices with random variables having absolutely continuous distributions with bounded densities. Under the hypothesis that the localization bounds hold for $0 < \alpha < \frac{1}{2}$, we prove that any nontrivial limit points of the random variables $\xi_{\omega,E}^N [I]$ are distributed according to Poisson distributions.

Matrix inequalities and majorizations around Hermite–Hadamard’s inequality

AbstractWe study the classical Hermite–Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such as the Schatten p-norm estimates $$ \begin{align*}\left(\|A^q\|_p^p + \|B^q\|_p^p\right)^{1/p} \le \|(xA+(1-x)B)^q\|_p+ \|((1-x)A+xB)^q\|_p, \end{align*} $$ for all positive (semidefinite) $n\times n$ matrices $A,B$ and $0&lt;q,x&lt;1$ . A related decomposition, with the assumption $X^*X+Y^*Y=XX^*+YY^*=I$ , is $$ \begin{align*}(X^*AX+Y^*BY)\oplus (Y^*AY+X^*BX) =\frac{1}{2n}\sum_{k=1}^{2n} U_k (A\oplus B)U_k^*, \end{align*} $$ for some family of $2n\times 2n$ unitary matrices $U_k$ . This is a majorization which is obtained by using the Hansen–Pedersen trace inequality.

Spectral theory for self-adjoint quadratic eigenvalue problems - a review

Many physical problems require the spectral analysis of quadratic matrix polynomials $M\lambda^2+D\lambda +K$, $\lambda \in \mathbb{C}$, with $n \times n$ Hermitian matrix coefficients, $M,\;D,\;K$. In this largely expository paper, we present and discuss canonical forms for these polynomials under the action of both congruence and similarity transformations of a linearization and also $\lambda$-dependent unitary similarity transformations of the polynomial itself. Canonical structures for these processes are clarified, with no restrictions on eigenvalue multiplicities. Thus, we bring together two lines of attack: (a) analytic via direct reduction of the $n \times n$ system itself by $\lambda$-dependent unitary similarity and (b) algebraic via reduction of $2n \times 2n$ symmetric linearizations of the system by either congruence (Section 4) or similarity (Sections 5 and 6) transformations which are independent of the parameter $\lambda$. Some new results are brought to light in the process. Complete descriptions of associated canonical structures (over $\mathbb{R}$ and over $\mathbb{C}$) are provided -- including the two cases of real symmetric coefficients and complex Hermitian coefficients. These canonical structures include the so-called sign characteristic. This notion appears in the literature with different meanings depending on the choice of canonical form. These sign characteristics are studied here and connections between them are clarified. In particular, we consider which of the linearizations reproduce the (intrinsic) signs associated with the analytic (Rellich) theory (Sections 7 and 9).

Real Zeros of Random Cosine Polynomials with Palindromic Blocks of Coefficients

It is well known that a random cosine polynomial \({V_n}\left(x \right) = \sum\nolimits_{j = 0}^n {{a_j}\cos \left({jx} \right)} \), x ∈ (0, 2π), with the coefficients being independent and identically distributed (i.i.d.) real-valued standard Gaussian random variables (asymptotically) has \(2n/\sqrt 3 \) expected real roots. On the other hand, out of many ways to construct a dependent random polynomial, one is to force the coefficients to be palindromic. Hence, it makes sense to ask how many real zeros a random cosine polynomial (of degree n) with identically and normally distributed coefficients possesses if the coefficients are sorted in palindromic blocks of a fixed length ℓ In this paper, we show that the asymptotics of the expected number of real roots of such a polynomial is \({{\rm{K}}_\ell} \times 2n/\sqrt 3 \), where the constant Kℓ (depending only on ℓ) is greater than 1, and can be explicitly represented by a double integral formula. That is to say, such polynomials have slightly more expected real zeros compared with the classical case with i.i.d. coefficients.

Riemannian Optimization on the Symplectic Stiefel Manifold

The symplectic Stiefel manifold, denoted by $\mathrm{Sp}(2p,2n)$, is the set of linear symplectic maps between the standard symplectic spaces $\mathbb{R}^{2p}$ and $\mathbb{R}^{2n}$. When $p=n$, it reduces to the well-known set of $2n\times 2n$ symplectic matrices. Optimization problems on $\mathrm{Sp}(2p,2n)$ find applications in various areas, such as optics, quantum physics, numerical linear algebra and model order reduction of dynamical systems. The purpose of this paper is to propose and analyze gradient-descent methods on $\mathrm{Sp}(2p,2n)$, where the notion of gradient stems from a Riemannian metric. We consider a novel Riemannian metric on $\mathrm{Sp}(2p,2n)$ akin to the canonical metric of the (standard) Stiefel manifold. In order to perform a feasible step along the antigradient, we develop two types of search strategies: one is based on quasi-geodesic curves, and the other one on the symplectic Cayley transform. The resulting optimization algorithms are proved to converge globally to critical points of the objective function. Numerical experiments illustrate the efficiency of the proposed methods.

Derivatives of symplectic eigenvalues and a Lidskii type theorem

AbstractAssociated with every $2n\times 2n$ real positive definite matrix $A,$ there exist n positive numbers called the symplectic eigenvalues of $A,$ and a basis of $\mathbb {R}^{2n}$ called the symplectic eigenbasis of A corresponding to these numbers. In this paper, we discuss differentiability and analyticity of the symplectic eigenvalues and corresponding symplectic eigenbasis and compute their derivatives. We then derive an analogue of Lidskii’s theorem for symplectic eigenvalues as an application.

Linear Equations Systems of Real and Complex Semi-Quaternions

In this work firstly real semi-quaternion matrices and their properties are examined. Then, $$2n\times 2n$$ complex adjoint matrix and $$4n\times 4n$$ real matrix representation of a real semi-quaternion matrix are expressed. Further systems of linear real semi-quaternionic equations are investigated and in some of them the $$2n\times 2n$$ complex adjoint matrix and the $$4n\times 4n$$ real matrix representation are used. Moreover, matrices which entries are complex semi-quaternions and their $$2n\times 2n$$ real semi-quaternion matrix representation are presented. Additionally system of linear complex semi-quaternionic equations is described with this $$2n\times 2n$$ matrix representation. Finally, for a complex semi-quaternion matrix $$4n\times 4n$$ complex adjoint matrix is introduced. Also, linear complex semi-quaternionic equations systems are examined.

Variational principles for symplectic eigenvalues

AbstractIf A is a real $2n \times 2n$ positive definite matrix, then there exists a symplectic matrix M such that $M^TAM=\text {diag}(D, D),$ where D is a positive diagonal matrix with diagonal entries $d_1(A)\leqslant \cdots \leqslant d_n(A).$ We prove a maxmin principle for $d_k(A)$ akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality $d_{i+j-1}(A+B)\geqslant d_i(A)+d_j(B).$

$$L^{p}-L^{p^{\prime }}$$ estimates for matrix Schrödinger equations

This paper is devoted to the study of dispersive estimates for matrix Schrödinger equations on the half-line with general boundary condition, and on the line. We prove $$L^{p}-L^{p^{\prime }}$$ estimates on the half-line for slowly decaying self-adjoint matrix potentials that satisfy $$\int _{0}^{\infty }\, (1+x) |V(x)|\, \mathrm{d}x < \infty $$ both in the generic and in the exceptional cases. We obtain our $$L^{p}-L^{p^{\prime }}$$ estimate on the line for a $$n \times n$$ system, under the condition that $$\int _{-^{\infty }}^{\infty }\, (1+|x|)\, |V(x)|\, \mathrm{d}x < \infty $$ , from the $$L^{p}-L^{p^{\prime }}$$ estimate for a $$2n\times 2n$$ system on the half-line. With our $$L^{p}-L^{p^{\prime }}$$ estimates, we prove Strichartz estimates.

Involutory-Multiple-Lightweight MDS Matrices based on Cauchy-type Matrices

&lt;p style='text-indent:20px;'&gt;One of the best methods for constructing maximum distance separable (&lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \operatorname{MDS} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;) matrices is based on making use of Cauchy matrices. In this paper, by using some extensions of Cauchy matrices, we introduce several new forms of &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \operatorname{MDS} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; matrices over finite fields of characteristic 2. A known extension of a Cauchy matrix, called the Cauchy-like matrix, with application in coding theory was introduced in 1985. One of the main contributions of this paper is to apply Cauchy-like matrices to introduce &lt;b&gt;&lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ 2n \times 2n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; involutory &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \operatorname{MDS} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; matrices&lt;/b&gt; in the semi-Hadamard form which is a generalization of the previously known methods. We make use of Cauchy-like matrices to construct &lt;b&gt;multiple &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \operatorname{MDS} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; matrices&lt;/b&gt; which can be used in the Feistel structures. We also introduce a new extension of Cauchy matrices to be referred to as &lt;i&gt;Cauchy-light matrices&lt;/i&gt;. The introduced Cauchy-light matrices are applied to construct &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ n \times n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ \operatorname{MDS} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; matrices having at least &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ 3n-3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; entries equal to the unit element &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ 1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;; such a matrix is called a &lt;b&gt;lightweight &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ \operatorname{MDS} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; matrix&lt;/b&gt; and can be used in the lightweight cryptography. A simple closed-form expression is given for the determinant of Cauchy-light matrices.&lt;/p&gt;

Optimization on the real symplectic group

We regard the real symplectic group $Sp(2n,\mathbb{R})$ as a constraint submanifold of the $2n\times 2n$ real matrices $\mathcal{M}_{2n}(\mathbb{R})$ endowed with the Euclidean (Frobenius) metric, respectively as a submanifold of the general linear group $Gl(2n,\mathbb{R})$ endowed with the (left) invariant metric. For a cost function that defines an optimization problem on the real symplectic group we give a necessary and sufficient condition for critical points and we apply this condition to the particular case of a least square cost function. In order to characterize the critical points we give a formula for the Hessian of a cost function defined on the real symplectic group, with respect to both considered metrics. For a generalized Brockett cost function we present a necessary condition and a sufficient condition for local minimum. We construct a retraction map that allows us to detail the steepest descent and embedded Newton algorithms for solving an optimization problem on the real symplectic group.

Elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces in $$\mathbf{R}^{2n}$$

Let $$\Sigma $$ be a compact convex hypersurface in $$\mathbf{R}^{2n}$$ which is P-cyclic symmetric, i.e., $$x\in \Sigma $$ implies $$Px\in \Sigma $$ with P being a $$2n\times 2n$$ symplectic orthogonal matrix and $$P^k=I_{2n}$$, where $$n, k\ge 2$$, $$ker(P-I_{2n})=0$$. In this paper, we first generalize Ekeland index theory for periodic solutions of convex Hamiltonian system to a index theory with P boundary value condition and study its relationship with Maslov P-index theory, then we use index theory to prove the existence of elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces in $$\mathbf{R}^{2n}$$ for a broad class of symplectic orthogonal matrix P.

Fingerprint Enhancement Based on Tensor of Wavelet Subbands for Classification

Fingerprint image enhancement is a key aspect of an automated fingerprint identification system. This paper describes an effective algorithm based on a novel lighting compensation scheme. The scheme involves the use of adaptive higher-order singular value decomposition on a tensor of wavelet subbands of a fingerprint (AHTWF) image to enhance the quality of the image. The algorithm consists of three stages. The first stage is the decomposition of an input fingerprint image of size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2M\times 2N$ </tex-math></inline-formula> into four subbands at the first level by applying a two-dimensional discrete wavelet transform. In the second stage, we construct a tensor in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {R}^{M\times N\times 4}$ </tex-math></inline-formula> space. The tensor contains four wavelet subbands that serve as four frontal planes. Furthermore, the tensor is decomposed through higher-order singular value decomposition to separate the fingerprint’s wavelet subbands into detailed individual components. In the third stage, a compensated image is produced by adaptively obtaining the compensation coefficient for each frontal plane of the tensor-based on the reference Gaussian template. The experimental results indicated that the quality of the AHTWF image was higher than that of the original image. The proposed algorithm not only improves the clarity and continuity of ridge structures but also removes the background and blurred regions of a fingerprint image. Therefore, this algorithm can achieve higher fingerprint classification accuracy than related methods can.

On 2-adic behavior of the number of domino tilings on torus

We study the 2-adic behavior of the number of domino tilings of a $2(2n+1)\times 2(2n+1)$ torus. We show that this number is of the form $2^{4n+3}g(n)^2+2^{8n+2}(2n+1)^{4n}h(n)$, where $g(n)$ and $h(n)$ are odd positive integers. Moreover, we prove that $g(n)$ and $h(n)$ are uniformly continuous under the 2-adic metric and invariant under interchanging $n$ and $-1-n$. This paper is an analogy of Henry Cohn's results for $2n\times 2n$ squares (Electron. J. Combin. 6 (1999)).

$${2\times 2}$$ 2 × 2 Matrix Representation Forms and Inner Relationships of Split Quaternions

This paper proposes a \({2\times 2}\) real matrix isomorphic representation form of split quaternions and a \({2m\times 2n}\) real matrix isomorphic representation form of \({m\times n}\) split quaternion matrices. In particular, we highlighted the inner relationships among the existing different \({2\times 2}\) matrix representation forms of split quaternions. Furthermore, we studied the inner relationships among different \({2\times 2}\) matrix representation forms of Hamilton quaternions as an application. The forms and relationships discussed in this paper can simplify the computation and make split quaternions get extensive applications.

A New Family of MRD Codes in &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$\mathbb{F_q}^{2n\times2n}$ &lt;/tex-math&gt; &lt;/inline-formula&gt; With Right and Middle Nuclei &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$\mathbb F_{q^n}$ &lt;/tex-math&gt; &lt;/inline-formula&gt;

In this paper, we present a new family of maximum rank distance (MRD for short) codes in $\mathbb F_{q}^{2n\times 2n}$ of minimum distance $2\leq d\leq 2n$. In particular, when $d=2n$, we can show that the corresponding semifield is exactly a Hughes-Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to $\mathbb F_{q^n}$. We also prove that the MRD codes of minimum distance $2<d<2n$ in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.