Positive Definite Research Articles

Ridge-type covariance and precision matrix estimators of the multivariate normal distribution

AbstractWe consider ridge-type estimation of the multivariate normal distribution’s covariance matrix and its inverse, the precision matrix. While several ridge-type covariance and precision matrix estimators have been presented in the literature, their respective inverses are often not considered as precision and covariance matrix estimators even though their estimands are one-to-one related through the matrix inverse. We study which estimator is to be preferred in what case. Hereto we compare the ridge-type covariance matrix estimators and their properties to that of the inverse of the ridge-type precision matrix estimators, and vice versa. The comparison, in which we take all ridge-type estimators along, is limited to a specific case that is illustrative of the difference between the covariance and precision matrix estimators. The comparison addresses the estimators’ estimating equation, analytic expression, analytic properties like positive definiteness and penalization limit, mean squared error, consistency, Bayesian formulation, and their loss and potential for marginal and partial correlation screening.

Theta Pairing of Hypersurface Rings

In this article, we prove a conjecture on the positive definiteness of the Hochster Theta pairing over a general isolated hypersurface singularity, namely: Let R be an admissible isolated hypersurface singularity of dimension n. If n is odd, then is positive semi-definite on . The conjecture is expected to be true for the polynomial ring over any field. We prove this conjecture over any field of arbitrary characteristic. We also provide two different proofs of the above conjecture overusing the Hodge theory of isolated hypersurface singularities and structural facts about the category of matrix factorizations. The first proof overis a more complete and developed version of a former work of the author. We have extended some of the former results in this article. The second proof overis quite direct and uses a former result of the author on Riemann-Hodge bilinear relations for Grothendieck residue pairing of isolated hypersurface singularities.

On Locally Symmetric Polynomial Metrics: Riemannian and Finslerian surfaces

In the paper we investigate locally symmetric polynomial metrics in special cases of Riemannian and Finslerian surfaces. The Riemannian case will be presented by a collection of basic results (regularity of second root metrics) and formulas up to Gauss curvature. In case of Finslerian surfaces we formulate necessary and sufficient conditions for a locally symmetric fourth root metric in 2D to be positive definite. They are given in terms of the coefficients of the polynomial metric to make checking the positive definiteness as simple and direct as possible. Explicit examples are also presented. The situation is more complicated in case of spaces of dimension more than two. Some necessary conditions and an explicit example are given for a positive definite locally symmetric polynomial metric in 3D. Computations are supported by the MAPLE mathematics software (LinearAlgebra).

A Switching Memory-Based Event-Trigger Scheme for Synchronization of Lur'e Systems With Actuator Saturation: A Hybrid Lyapunov Method.

This article is concerned with the event-triggered synchronization of Lur'e systems subject to actuator saturation. Aiming at reducing control costs, a switching-memory-based event-trigger (SMBET) scheme, which allows a switching between the sleeping interval and the memory-based event-trigger (MBET) interval, is first presented. In consideration of the characteristics of SMBET, a piecewise-defined but continuous looped-functional is newly constructed, under which the requirement of positive definiteness and symmetry on some Lyapunov matrices is dropped within the sleeping interval. Then, a hybrid Lyapunov method (HLM), which bridges the gap between the continuous-time Lyapunov theory (CTLT) and the discrete-time Lyapunov theory (DTLT), is used to make the local stability analysis of the closed-loop system. Meanwhile, using a combination of inequality estimation techniques and the generalized sector condition, two sufficient local synchronization criteria and a codesign algorithm for the controller gain and triggering matrix are developed. Furthermore, two optimization strategies are, respectively, put forward to enlarge the estimated domain of attraction (DoA) and the allowable upper bound of sleeping intervals on the premise of ensuring local synchronization. Finally, a three-neuron neural network and the classical Chua's circuit are used to carry out some comparison analyses and to display the advantages of the designed SMBET strategy and the constructed HLM, respectively. Also, an application to image encryption is provided to substantiate the feasibility of the obtained local synchronization results.

Denoising of imaginary time response functions with Hankel projections

Imaginary-time response functions of finite-temperature quantum systems are often obtained with methods that exhibit stochastic or systematic errors. Reducing these errors comes at a large computational cost—in quantum Monte Carlo simulations, the reduction of noise by a factor of two incurs a simulation cost of a factor of four. In this paper, we relate certain imaginary-time response functions to an inner product on the space of linear operators on Fock space. We then show that data with noise typically does not respect the positive definiteness of its associated Gramian. The Gramian has the structure of a Hankel matrix. As a method for denoising noisy data, we introduce an alternating projection algorithm that finds the closest positive definite Hankel matrix consistent with noisy data. We test our methodology at the example of fermion Green's functions for continuous-time quantum Monte Carlo data and show remarkable improvements of the error, reducing noise by a factor of up to 20 in practical examples. We argue that Hankel projections should be used whenever finite-temperature imaginary-time data of response functions with errors is analyzed, be it in the context of quantum Monte Carlo, quantum computing, or in approximate semianalytic methodologies. Published by the American Physical Society 2024

Formation control of high-order linear discrete-time multiple unmanned aerial vehicles systems in noisy environments

Formation control of high-order linear discrete-time multiple unmanned aerial vehicles (multi-UAVs) systems with multiplicative noises is studied in this article. Because of the existence of multiplicative noises, classical methods for discrete-time system control problems should be improved. Here, we develop a new discrete-time formation protocol for multi-UAVs systems. The stochastic Lyapunov stability theory and the positive definiteness theory of a generalized algebraic Riccati equation are then used to derive sufficient conditions for formation feasibility.

An improved preconditioned conjugate gradient method for unconstrained optimization problem with application in Robot arm control

AbstractThis work suggests improved conjugate gradient methods for enhancing the efficiency and robustness of the classical conjugate gradient methods. The study modifies the diagonal of the inverse Hessian approximation of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi‐Newton update in order to build a preconditioner for nonlinear conjugate gradient (NCG) methods applied to large‐scale unconstrained optimization problems. Damping techniques were embedded into the algorithm to impose the positive definiteness of the diagonal approximation. This made the methods easy to use and presented a viable way to increase the effectiveness of unconstrained optimization techniques. Experimental findings from a collection of benchmark problems demonstrate the efficiency and robustness of the proposed method when compared to five other existing NCG algorithms. Moreover, the successful application of the algorithm to manipulate robotic planar motion control systems with 3 degrees of freedom has been demonstrated, highlighting the practicality of the proposed approach.

Direct estimation and inference of higher-level correlations from lower-level measurements with applications in gene-pathway and proteomics studies.

This paper tackles the challenge of estimating correlations between higher-level biological variables (e.g. proteins and gene pathways) when only lower-level measurements are directly observed (e.g. peptides and individual genes). Existing methods typically aggregate lower-level data into higher-level variables and then estimate correlations based on the aggregated data. However, different data aggregation methods can yield varying correlation estimates as they target different higher-level quantities. Our solution is a latent factor model that directly estimates these higher-level correlations from lower-level data without the need for data aggregation. We further introduce a shrinkage estimator to ensure the positive definiteness and improve the accuracy of the estimated correlation matrix. Furthermore, we establish the asymptotic normality of our estimator, enabling efficient computation of P-values for the identification of significant correlations. The effectiveness of our approach is demonstrated through comprehensive simulations and the analysis of proteomics and gene expression datasets. We develop the R package highcor for implementing our method.

Approximation‐based implicit integration algorithm for the Simo‐Miehe model of finite‐strain inelasticity

AbstractWe propose a simple, efficient, and reliable procedure for implicit time stepping, regarding a special case of the viscoplasticity model proposed by Simo and Miehe (1992). The kinematics of this popular model is based on the multiplicative decomposition of the deformation gradient tensor, allowing for a combination of Newtonian viscosity and arbitrary isotropic hyperelasticity. The algorithm is based on approximation of precomputed solutions. Both Lagrangian and Eulerian versions of the algorithm with equivalent properties are available. The proposed numerical scheme is non‐iterative, unconditionally stable, and first order accurate. Moreover, the integration algorithm strictly preserves the inelastic incompressibility constraint, symmetry, positive definiteness, and w‐invariance. The accuracy of stress calculations is verified in a series of numerical tests, including non‐proportional loading and large strain increments. In terms of stress calculation accuracy, the proposed algorithm is equivalent to the implicit Euler method with strict inelastic incompressibility. The algorithm is implemented into MSC.MARC and a demonstration initial‐boundary value problem is solved.

Positive definiteness of Hadamard exponentials and Hadamard inverses

Positive definiteness of Hadamard exponentials and Hadamard inverses

Unconditional optimal [formula omitted]-norm error estimate and superconvergence analysis of a linearized nonconforming finite element variable-time-step BDF2 method for the nonlinear complex Ginzburg–Landau equation

In this paper, a linearized fully discrete scheme combining the variable-time-step two-step backward differentiation formula (VSBDF2) in time and the nonconforming finite element methods (FEMs) in space is constructed and analyzed for the nonlinear complex Ginzburg–Landau equation. A novel convergence analysis approach is proposed, which shows that the H1-norm error of this scheme can reach optimal order in space and sharp second-order convergence in time. The key ingredients of our analysis are temporal–spatial error splitting approach, a new positive definiteness property of discrete orthogonal convolution (DOC) kernels with respect to complex sequences, an energy projection operator and a discrete Laplace operator Δh related to the nonconforming element space, and a relationship between Δh and classical L2 projection operator in complex space. Furthermore, through a specific treatment of the error equation and the combination technique of interpolating operator and projection operator, we prove that the proposed scheme has the global superconvergence result of order O(τ2+h2) in H1-norm for the first time, which further improves the efficiency of numerical computation. Here τ and h are the maximum time and spatial step sizes, respectively. It should be pointed out that all these error results are rigorously established under mild assumptions of the time step, and get rid of the usual grid ratio constraints between temporal and spatial step sizes in linearized scheme. At last, numerical experiments are provided to verify the theoretical analysis.

Visibility graph-based covariance functions for scalable spatial analysis in non-convex partially Euclidean domains.

We present a new method for constructing valid covariance functions of Gaussian processes for spatial analysis in irregular, non-convex domains such as bodies of water. Standard covariance functions based on geodesic distances are not guaranteed to be positive definite on such domains, while existing non-Euclidean approaches fail to respect the partially Euclidean nature of these domains where the geodesic distance agrees with the Euclidean distances for some pairs of points. Using a visibility graph on the domain, we propose a class of covariance functions that preserve Euclidean-based covariances between points that are connected in the domain while incorporating the non-convex geometry of the domain via conditional independence relationships. We show that the proposed method preserves the partially Euclidean nature of the intrinsic geometry on the domain while maintaining validity (positive definiteness) and marginal stationarity of the covariance function over the entire parameter space, properties which are not always fulfilled by existing approaches to construct covariance functions on non-convex domains. We provide useful approximations to improve computational efficiency, resulting in a scalable algorithm. We compare the performance of our method with those of competing state-of-the-art methods using simulation studies on synthetic non-convex domains. The method is applied to data regarding acidity levels in the Chesapeake Bay, showing its potential for ecological monitoring in real-world spatial applications on irregular domains.

Differentiable microstructures design via anisotropic thermal diffusion

We propose a novel method to design differentiable microstructures. Central to our algorithm is a new representation of the mapping from the parameters to microstructures, formulated as the anisotropic thermal diffusion. A metric field governs the anisotropic diffusion. The metric associated with each point is represented as a 2 × 2 symmetric positive definite matrix that becomes the design variable. To alleviate the difficulties caused by symmetric positive definite constraints, we perform the singular value decomposition of the metric matrix so that the design variable includes a rotation angle and a diagonal matrix. Then, the positive definiteness is converted to requiring the two diagonal entries of the diagonal matrix to be positive, which is easier to deal with. The effectiveness of our algorithm is demonstrated through evaluations and comparisons over various examples.

Ghost problem, spectrum identities and various constraints on brane-localized gravity

This paper investigates the brane-localized interactions, including DGP gravity and higher derivative (HD) gravity localized on the brane. We derive the effective action on the brane, which suggests the brane-localized HD gravity suffers the ghost problem generally. Besides, we obtain novel algebraic identities of the mass spectrum, which reveal the global nature and can characterize the phase transformation of the mass spectrum. We get a powerful ghost-free condition from the spectrum identities, which rules out one type of brane-localized HD gravity. We further prove the mass spectrum is real and non-negative m2 ≥ 0 under the ghost-free condition.Furthermore, we discuss various constraints on parameters of brane-localized gravity in AdS/BCFT and wedge holography, respectively. They include the ghost-free condition of Kaluza-Klein and brane-bending modes, the positive definiteness of boundary central charges, and entanglement entropy. The ghost-free condition imposes strict constraint, which requires non-negative couplings for pure DGP gravity and Gauss-Bonnet gravity on the brane. It also rules out one class of brane-localized HD gravity. Thus, such HD gravity should be understood as a low-energy effective theory on the brane under the ghost energy scale. Finally, we briefly discuss the applications of our results.

Block-diagonal idiosyncratic covariance estimation in high-dimensional factor models for financial time series

Estimation of high-dimensional covariance matrices in latent factor models is an important topic in many fields and especially in finance. Since the number of financial assets grows while the estimation window length remains of limited size, the often used sample estimator yields noisy estimates which are not even positive definite. Under the assumption of latent factor models, the covariance matrix is decomposed into a common low-rank component and a full-rank idiosyncratic component. In this paper we focus on the estimation of the idiosyncratic component, under the assumption of a grouped structure of the time series, which may arise due to specific factors such as industries, asset classes or countries. We propose a generalized methodology for estimation of the block-diagonal idiosyncratic component by clustering the residual series and applying shrinkage to the obtained blocks in order to ensure positive definiteness. We derive two different estimators based on different clustering methods and test their performance using simulation and historical data. The proposed methods are shown to provide reliable estimates and outperform other state-of-the-art estimators based on thresholding methods.

A novel equivalent reciprocal convex combination technique to sampled‐data master–slave synchronization for chaotic Lur'e systems with time‐varying delays

AbstractFor a class of master–slave (M‐S) systems in chaotic Lur'e systems with time‐varying delays, a sampled‐data synchronization controller is designed, and a new synchronization stability condition is proposed in the form of linear matrix inequality. A novel Lyapunov‐Krasovskii functional (LKF) is constructed by using the looped‐functional method, and the positive definiteness condition of LKF including sampled‐data parts is exchanged with a looping condition by constructing a functional, which should be equal at adjacent sampling times. The M‐S synchronization condition is obtained utilizing the equivalent reciprocal convex combination approach combined with Bessel‐Legendre integral inequality to estimate the LKF derivative. Different from previous methods, due to fully utilizing both nonlinearity and state information at the sampling time via the integral of error systems from the sampling time to current one, the M‐S synchronization condition is less conservative, and obtained sampled‐data controller possesses a longer sampling period. Finally, two numerical examples verify the superiority and validity of the approach.

Numerical algorithm for estimating a conditioned symmetric positive definite matrix under constraints

SummaryWe present RCO (regularized Cholesky optimization): a numerical algorithm for finding a symmetric positive definite (PD) matrix with a bounded condition number that minimizes an objective function. This task arises when estimating a covariance matrix from noisy data or due to model constraints, which can cause spurious small negative eigenvalues. A special case is the problem of finding the nearest well‐conditioned PD matrix to a given matrix. RCO explicitly optimizes the entries of the Cholesky factor. This requires solving a regularized non‐linear, non‐convex optimization problem, for which we apply Newton‐CG and exploit the Hessian's sparsity. The regularization parameter is determined via numerical continuation with an accuracy‐conditioning trade‐off criterion. We apply RCO to our motivating educational measurement application of estimating the covariance matrix of an empirical best linear prediction (EBLP) of school growth scores. We present numerical results for two empirical datasets, state and urban. RCO outperforms general‐purpose near‐PD algorithms, obtaining ‐smaller matrix reconstruction bias and smaller EBLP estimator mean‐squared error. It is in fact the only algorithm that solves the right minimization problem, which strikes a balance between the objective function and the condition number. RCO can be similarly applied to the stable estimation of other posterior means. For the task of finding the nearest PD matrix, RCO yields similar condition numbers to near‐PD methods, but provides a better overall near‐null space.

Innovative launch collision avoidance (LCOLA) tool prioritizing accuracy, launch access and efficiency

A new Launch Collision Avoidance (LCOLA) analysis capability is presented. The method represents the combination of the patented algorithm developed by Analytical Graphics Inc. and implemented in its Launch Window Analysis (LWA) tool, together with a new LCOLA collision probability algorithm developed by COMSPOC Corporation to maximize launch window using responsive, comprehensive, and efficient LCOLA closure windows. The new tool works by assessing the underlying topography of the launch closure problem to identify and characterize all launch holds within the user-specified launch window rapidly and accurately without confining the launch operator to “top-of-the-minute” or other such constructs.This new LCOLA capability avoids the pitfalls of the typical “digitized” or “snapshot” launch window screening approach, where launch objects are compared against the on-orbit catalogue at a small launch “T-zero” step size.The new capability can detect launch closure holds based upon miss distance, collision probability or both. Prior to the probability calculation, each covariance is tested for positive definiteness and remediated if necessary. The tool's support for parallel processing, combined with its algorithmic efficiency, allow it to process all deployed objects. The tool aggregates all resulting launch holds into a concise report for the launch director and team.

Imitating via manipulability: Geometry-aware combined DMP with via-point and speed adaptation

Manipulability ellipsoid on the Riemannian manifold provides an effective criterion for guiding the regulation of robot postures in an efficient and natural manner. While many manipulability-based learning by demonstration (LbD) methods achieve favorable performance in human-like manipulation, the adaptation of movement skills encapsulated in Symmetric Positive Definite (SPD) matrices and the implementation of manipulability-relevant skills related to velocity scaling are still largely open. In this paper, we develop a new framework based on Geometry-Aware Combined Dynamic Movement Primitives (GA-CDMP) for learning movement skills from demonstrations. The GA-CDMP model not only adapts learned trajectories to SPD-via-points but also establishes a correlation between SPD-based trajectories and position trajectories, enabling the adaptation of learned SPD-based trajectories across different non-linear motion velocity scales. The experimental evaluations show that our proposed method exhibits remarkable precision in passing through via-points positioned at a distance from the demonstration, while also substantially improving the invariant shape similarity in manipulability by approximately 65% compared to extended Kernelized Movement Primitives (KMPs). Moreover, our velocity adaptive approach achieves impressive success rates of up to 94% and demonstrates a notable reduction in execution time (11.72 ± 0.45 s), representing an almost 10% decrease in comparison to the state-of-the-art method employed in water-carrying tasks.

An inexact variable metric variant of incremental aggregated Forward-Backward method for the large-scale composite optimization problem

ABSTRACT This paper focuses on the large-scale composite optimization problem, which is the sum of a finite number of smooth (possibly nonconvex) functions and a convex (possibly nonsmooth) function. A common method to solve this problem is proximal incremental aggregated gradient (PIAG) method (N. D. Vanli, M. Gurbuzbalaban and A. Ozdaglar, SIAM Journal on Optimization, 2018, 28(2): 1282–1300), which exploits the gradient information from previous iterations to approximate the gradient at the current point. However, as many first-order algorithms, it may suffer from slow convergence. To address this issue, this paper proposes an inexact variable metric variant of incremental aggregated Forward-Backward (iVMFB-IAG) method. This methods incorporates a variable metric strategy to accelerate the PIAG method and introduces an inexact criterion to enhance its practicality. Under the Kurdyka-Łojasiewicz (KL) property and the uniformly bounded positive definiteness of the scaling matrix, the sequence generated by the iVMFB-IAG method converges globally to the critical point of the problem. Additionally, the convergence rate is established under the KL framework. Moreover, under certain restrictive conditions, the method exhibits local superlinear convergence. Through the numerical experiments on image reconstruction applications, we demonstrate the effectiveness of the iVMFB-IAG method by restoring two distinct images.