scholarly journals Accelerated Diffusion-Based Sampling by the Non-Reversible Dynamics with Skew-Symmetric Matrices

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 993
Author(s):  
Futoshi Futami ◽  
Tomoharu Iwata ◽  
Naonori Ueda ◽  
Issei Sato
Keyword(s):  
Symmetric Matrix ◽  
Hessian Matrix ◽  
Sampling Technique ◽  
Langevin Dynamics ◽  
Superior Performance ◽  
Symmetric Matrices ◽  
Theoretical Understanding ◽  
Large Memory ◽  
Practical Algorithm ◽  
Memory Efficient

Langevin dynamics (LD) has been extensively studied theoretically and practically as a basic sampling technique. Recently, the incorporation of non-reversible dynamics into LD is attracting attention because it accelerates the mixing speed of LD. Popular choices for non-reversible dynamics include underdamped Langevin dynamics (ULD), which uses second-order dynamics and perturbations with skew-symmetric matrices. Although ULD has been widely used in practice, the application of skew acceleration is limited although it is expected to show superior performance theoretically. Current work lacks a theoretical understanding of issues that are important to practitioners, including the selection criteria for skew-symmetric matrices, quantitative evaluations of acceleration, and the large memory cost of storing skew matrices. In this study, we theoretically and numerically clarify these problems by analyzing acceleration focusing on how the skew-symmetric matrix perturbs the Hessian matrix of potential functions. We also present a practical algorithm that accelerates the standard LD and ULD, which uses novel memory-efficient skew-symmetric matrices under parallel-chain Monte Carlo settings.

scholarly journals The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

2019 ◽  
Vol 7 (1) ◽  
pp. 257-262
Author(s):  
Kenji Toyonaga
Keyword(s):  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Multiplicity ◽  
Necessary And Sufficient ◽  
Multiplicity Of An Eigenvalue

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

scholarly journals A New Filter Nonmonotone Adaptive Trust Region Method for Unconstrained Optimization

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 208 ◽  
Author(s):  
Xinyi Wang ◽  
Xianfeng Ding ◽  
Quan Qu
Keyword(s):  
Unconstrained Optimization ◽  
Symmetric Matrix ◽  
Hessian Matrix ◽  
Trust Region Method ◽  
Trust Region ◽  
Adaptive Strategy ◽  
Step Length ◽  
Computational Costs ◽  
Definite Symmetric Matrix ◽  
Approximate Hessian

In this paper, a new filter nonmonotone adaptive trust region with fixed step length for unconstrained optimization is proposed. The trust region radius adopts a new adaptive strategy to overcome additional computational costs at each iteration. A new nonmonotone trust region ratio is introduced. When a trial step is not successful, a multidimensional filter is employed to increase the possibility of the trial step being accepted. If the trial step is still not accepted by the filter set, it is possible to find a new iteration point along the trial step and the step length is computed by a fixed formula. The positive definite symmetric matrix of the approximate Hessian matrix is updated using the MBFGS method. The global convergence and superlinear convergence of the proposed algorithm is proven by some classical assumptions. The efficiency of the algorithm is tested by numerical results.

At Risk of “Leftover Singles”: Dimensions and Sociopsychological Repercussions of Delayed Marriageability Among Educated Females in Pakistan

2020 ◽  
Vol 28 (4) ◽  
pp. 403-412
Author(s):  
Tipu Sultan ◽  
Saeed Ahmad ◽  
Ayesha Ayub
Keyword(s):  
Marriage Market ◽  
Sampling Technique ◽  
Caste System ◽  
Sociocultural Factors ◽  
Snowball Sampling ◽  
Theoretical Understanding ◽  
Personal Traits ◽  
Interview Guide ◽  
Family Based

The current study examined the educational, economic, and sociocultural, family and personal dimensions of delayed marriage among educated females in Pakistan. In addition, it revealed family-based and personal dimensions of delayed marriage among educated females in Pakistan. Furthermore, it highlighted the sociopsychological consequences of delayed marriageability among females in patriarchal society. For this purpose, 35 females, in the age bracket of 30–49 years and with a minimum of 16 years of education, were recruited for the current study through purposive and snowball sampling technique. An interview guide was used as a tool for data collection. The main sociocultural factors of delayed marriageability were the unavailability of a suitable match in the marriage market, the provision of the dowry, the pivotal role of the caste system and the second fiddle role of sectarian affiliation, and previous marital status (engaged or divorced) of the females. The structure and the size of the family were also the decisive family factors of delayed marriageability. Additionally, among personal traits, physical outlook and effective individualism played a prominent role. The current study conjectured a relational insight and transformation in a family structure for family demographers. It was the first qualitative study to highlight the patriarchal perspective of Pakistani society on the factors of delayed marriages. The findings of the current study would enrich the overall theoretical understanding of delayed marriageability among females.

scholarly journals On Turnbull identity for skew-symmetric matrices

2000 ◽  
Vol 43 (2) ◽  
pp. 379-393 ◽  
Author(s):  
Tôru Umeda ◽  
Takeshi Hirai
Keyword(s):  
Differential Operators ◽  
Symmetric Matrix ◽  
Point Of View ◽  
Symmetric Matrices ◽  
Invariant Differential Operators ◽  
Type Identity ◽  
Central Elements ◽  
Theoretic Point ◽  
Invariant Differential ◽  
Matrix Spaces

AbstractIn the last six lines of Turnbull's 1948 paper, he left an enigmatic statement on a Capelli-type identity for skew-symmetric matrix spaces. In the present paper, on Turnbull's suggestion, we show that certain Capelli-type identities hold for this case. Our formulae connect explicitly the central elements inU(gln) to the invariant differential operators, both of which are expressed with permanent. This also clarifies the meaning of Turnbull's statement from the Lie-theoretic point of view.

Recovery of Low Rank Symmetric Matrices via Schatten p Norm Minimization

2016 ◽  
Vol 33 (01) ◽  
pp. 1650003
Author(s):  
Li Cui ◽  
Lu Liu ◽  
Di-Rong Chen ◽  
Jian-Feng Xie
Keyword(s):  
Symmetric Matrix ◽  
Linear Measurement ◽  
Low Rank ◽  
Symmetric Matrices ◽  
Norm Minimization ◽  
Rank Matrix ◽  
Matrix Formula ◽  
Low Rank Matrix Recovery ◽  
Low Rank Matrix ◽  
Recovery Problem

In this paper, we give an application of the perturbation inequality to the low rank matrix recovery problem and provide a condition on the linear map of underdetermined linear system that every minimal rank symmetric matrix [Formula: see text] can be exactly recovered from the linear measurement [Formula: see text] via some Schatten [Formula: see text] norm minimization. Moreover it is shown that the explicit bound on exponent [Formula: see text] in the Schatten [Formula: see text] norm minimization can be exactly extracted.

LU-FACTORIZATIONS OF SYMMETRIC MATRICES WITH APPLICATIONS

2010 ◽  
Vol 03 (01) ◽  
pp. 133-143 ◽  
Author(s):  
Guoyou Qian ◽  
Jingya Lu
Keyword(s):  
Symmetric Matrix ◽  
Symmetric Matrices ◽  
Lu Factorization

In this paper, we describe explicitly the LU -factorization of a symmetric matrix of order n with n ≤ 7 when each of its ordered principal minors is nonzero. By using this result and some other related results on non-singularity previously given by Smith, Beslin, Hong, Lee and Ligh in the literature, we establish several theorems concerning LU -factorizations of power GCD matrices, power LCM matrices and reciprocal power GCD matrices and reciprocal power LCM matrices.

scholarly journals A special property of the matrix Riccati equation

1974 ◽  
Vol 10 (2) ◽  
pp. 245-253 ◽  
Author(s):  
A.N. Stokes
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Symmetric Matrix ◽  
Special Property ◽  
Positive Definiteness ◽  
Symmetric Matrices ◽  
Unusual Property ◽  
Riccati Differential Equation ◽  
The Matrix ◽  
Independent Variable

In the domain of real symmetric matrices ordered by the positive definiteness criterion, the symmetric matrix Riccati differential equation has the unusual property of preserving the ordering of its solutions as the independent variable changes, Here is is shown that, subject to a continuity restriction, the Riccati equation is unique among comparable equations in possessing this property.

Cayley formula in Minkowski space-time

2015 ◽  
Vol 12 (05) ◽  
pp. 1550058 ◽  
Author(s):  
Melek Erdoğdu ◽  
Mustafa Özdemir
Keyword(s):  
Minkowski Space ◽  
Symmetric Matrix ◽  
Space Time ◽  
Rotation Matrix ◽  
Symmetric Matrices ◽  
Rotation Matrices ◽  
Minkowski Space Time ◽  
Real Matrices ◽  
Skew Symmetric Matrix

In this paper, Cayley formula is derived for 4 × 4 semi-skew-symmetric real matrices in [Formula: see text]. For this purpose, we use the decomposition of a semi-skew-symmetric matrix A = θ1A1 + θ2A2 by two unique semi-skew-symmetric matrices A1 and A2 satisfying the properties [Formula: see text] and [Formula: see text] Then, we find Lorentzian rotation matrices with semi-skew-symmetric matrices by Cayley formula. Furthermore, we give a way to find the semi-skew-symmetric matrix A for a given Lorentzian rotation matrix R such that R = Cay (A).

scholarly journals On the smallest singular value of symmetric random matrices

2021 ◽  
pp. 1-22
Author(s):  
Vishesh Jain ◽  
Ashwin Sah ◽  
Mehtaab Sawhney
Keyword(s):  
Random Matrices ◽  
Symmetric Matrix ◽  
Geometric Approach ◽  
Random Variable ◽  
Symmetric Matrices ◽  
Common Denominator ◽  
Geometric Framework ◽  
Small Constant ◽  
Special Case ◽  
Smallest Singular Value

Abstract We show that for an $n\times n$ random symmetric matrix $A_n$ , whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $\xi$ with mean 0 and variance 1, \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O_{\xi}(\epsilon^{1/8} + \exp(\!-\Omega_{\xi}(n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} This improves a result of Vershynin, who obtained such a bound with $n^{1/2}$ replaced by $n^{c}$ for a small constant c, and $1/8$ replaced by $(1/8) - \eta$ (with implicit constants also depending on $\eta > 0$ ). Furthermore, when $\xi$ is a Rademacher random variable, we prove that \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O(\epsilon^{1/8} + \exp(\!-\Omega((\!\log{n})^{1/4}n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} The special case $\epsilon = 0$ improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that $\mathbb{P}[s_n(A_n) = 0] \le O(\exp(\!-\Omega(n^{1/2}))).$ Notably, in a departure from the previous two best bounds on the probability of singularity of symmetric matrices, which had relied on somewhat specialized and involved combinatorial techniques, our methods fall squarely within the broad geometric framework pioneered by Rudelson and Vershynin, and suggest the possibility of a principled geometric approach to the study of the singular spectrum of symmetric random matrices. The main innovations in our work are new notions of arithmetic structure – the Median Regularized Least Common Denominator (MRLCD) and the Median Threshold, which are natural refinements of the Regularized Least Common Denominator (RLCD)introduced by Vershynin, and should be more generally useful in contexts where one needs to combine anticoncentration information of different parts of a vector.

scholarly journals A note on multilevel Toeplitz matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 114-126
Author(s):  
Lei Cao ◽  
Selcuk Koyuncu
Keyword(s):  
Toeplitz Matrix ◽  
Symmetric Matrix ◽  
Toeplitz Matrices ◽  
Tensor Products ◽  
Symmetric Matrices ◽  
Unitary Matrices ◽  
Complex Symmetric Matrix ◽  
Complex Symmetric ◽  
Complex Symmetric Matrices

Abstract Chien, Liu, Nakazato and Tam proved that all n × n classical Toeplitz matrices (one-level Toeplitz matrices) are unitarily similar to complex symmetric matrices via two types of unitary matrices and the type of the unitary matrices only depends on the parity of n. In this paper we extend their result to multilevel Toeplitz matrices that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. We provide a method to construct the unitary matrices that uniformly turn any multilevel Toeplitz matrix to a complex symmetric matrix by taking tensor products of these two types of unitary matrices for one-level Toeplitz matrices according to the parity of each level of the multilevel Toeplitz matrices. In addition, we introduce a class of complex symmetric matrices that are unitarily similar to some p-level Toeplitz matrices.

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