Trace Of Matrix Research Articles

On the Optimality of Sufficient Statistics-Based Quantizers.

Let X be a random variable taking values in a set X, and let {Pθ; θ ∈ Θ} be a family of distributions indexed by the parameter vector θ taking values in a set Θ. A quantized random variable γ(X) is obtained by employing a quantizer γ: X→ {1,…,K}. It is shown that any extreme point of the set of all possible probability distributions of γ(X) can be achieved by a deterministic quantizer that decides based only on the sufficient statistics. Using this fact, optimality properties of deterministic sufficient statistics-based quantizers are established for the problem of parameter estimation. It is proven that there always exists an optimal partitioning of sufficient statistics into K convex polytopes which maximizes the trace of the Fisher information matrix when {Pθ; θ ∈ Θ} belongs to the exponential family. Furthermore, the optimality of likelihood ratio statistic for simple hypothesis testing follows as a consequence of this result, thereby demonstrating a link between parameter estimation and hypothesis testing.

Illinois-Type Methods for Noisy Euclidean Distance Realization

In this work, we introduce an iterative algorithm for the Euclidean distance matrix completion (EDMC) problem with noisy and incomplete distance measurements. The proposed method is based on semidefinite programming, utilizes a Pareto iterative approach, and performs a projection-free convex optimization over the spectrahedron to solve a level-set problem relevant to EDMC problems. The optimality trade-off between the trace of a positive semidefinite matrix and a loss function is pursued over Pareto optimal points with simple, derivative-free, costly efficient nonlinear equation root finding iterations called Illinois-type methods. We evaluate our approach numerically in a scenario where distance measurements are affected by multiplicative noise.

Array Orientation Adjustments Subject to Optimal Direct Position Determination Performance

Arrays deployed in antenna-array-based direct position determination (DPD) systems are conventionally placed parallel to each other with invariant orientations, which may be inappropriate. This paper proposes a novel array orientation adjustment method to achieve optimal DPD performance, in which we seek optimal array orientations that maximize the trace of the Fisher information matrix (FIM) and then minimize localization uncertainty. The optimization problem is unconstrained, multivariable, and can be converted into multiple tractable single-variable optimization problems. An exact solution can be obtained by a straightforward but computationally tedious one-dimensional exhaustive search. To reduce the computational load, we then derive an asymptotic solution in cases where the arrays consist of massive antennas. Numerical examples demonstrate the effectiveness and feasibility of the proposed method in resolution and precision enhancement, especially in a low signal-to-noise ratio (SNR).

A Hybrid Approach to Optimal TOA-Sensor Placement With Fixed Shared Sensors for Simultaneous Multi-Target Localization

This paper focuses on optimal time-of-arrival (TOA) sensor placement for multiple target localization simultaneously. In previous work, different solutions only using non-shared sensors to localize multiple targets have been developed. Those methods localize different targets one-by-one or use a large number of mobile sensors with many limitations, such as low effectiveness and high network complexity. In this paper, firstly, a novel optimization model for multi-target localization incorporating shared sensors is formulated. Secondly, the systematic theoretical results of the optimal sensor placement are derived and concluded using the A-optimality criterion, i.e., minimizing the trace of the inverse Fisher information matrix (FIM), based on rigorous geometrical derivations. The reachable optimal trace of Cramér-Rao lower bound (CRLB) is also derived. It can provide optimal conditions for many cases and even closed form solutions for some special cases. Thirdly, a novel numerical optimization algorithm to quickly find and calculate the (sub-)optimal placement and achievable lower bound is explored, when the model becomes complicated with more practical constraints. Then, a hybrid method for solving the most general situation, integrating both the analytical and numerical solutions, is proposed. Finally, the correctness and effectiveness of the proposed theoretical and mathematical methods are demonstrated by several simulation examples.

TCVQ-SVM algorithm for narrowband spectrum sensing

Spectrum sensing is viewed as the basic and crucial technology for cognitive radio. To improve the accuracy of spectrum sensing in low signal to noise ratio (SNR), this paper presents an efficient TCVQ-SVM method based on machine learning for narrowband spectrum sensing. Firstly, trace of covariance matrix and variance of quadratic covariance matrix (TCVQ) is extracted as feature vectors and combined as training samples of spectrum sensing. Then, the classification model can be achieved by training samples based on support vector machine (SVM), which can avoid setting threshold and adjusting classification hyperplane by its self-learning ability. Lastly, the result of spectrum sensing can be obtained. By utilizing trace and variance as input features of SVM, the algorithm can make full use of the eigenvalue difference and structure characteristic of the received signal, and at the same time, achieve good performance in low SNR. Theoretical analysis reveals that the proposed method has low computational complexity. Simulation results and experiments on the hardware platform illustrate that the proposed algorithm is effective and robust.

Properties of Fisher information gain for Bayesian design of experiments

The Bayesian decision-theoretic approach to design of experiments involves specifying a design (values of all controllable variables) to maximise the expected utility function (expectation with respect to the distribution of responses and parameters). For most common utility functions, the expected utility is rarely available in closed form and requires a computationally expensive approximation which then needs to be maximised over the space of all possible designs. This hinders practical use of the Bayesian approach to find experimental designs. However, recently, a new utility called Fisher information gain has been proposed. The resulting expected Fisher information gain reduces to the prior expectation of the trace of the Fisher information matrix. Since the Fisher information is often available in closed form, this significantly simplifies approximation and subsequent identification of optimal designs. In this paper, it is shown that for exponential family models, maximising the expected Fisher information gain is equivalent to maximising an alternative objective function over a reduced-dimension space, simplifying even further the identification of optimal designs. However, if this function does not have enough global maxima, then designs that maximise the expected Fisher information gain lead to non-identifiability.

Estimating Number of Critical Eigenvalues of Large-Scale Power System Based on Contour Integral

This paper proposes an improved stochastic estimation method based on contour integral to estimate the number of critical eigenvalues. The stochastic estimation method transforms the eigenvalue numbers' calculation into summing the traces of a series of inverse matrices. And then, a stochastic strategy is used to estimate the trace of the inverse matrix. However, the method results in an unacceptable error when the state matrices of many practical power systems are ill-conditioned. To solve this problem, the improved algorithm is proposed by formulating a satisfactory matrix for trace calculation artificially instead of randomly with the help of the well-known greedy algorithm based on graph coloring. It is proved to be much more reliable when dealing with ill matrix. The proposed method based on contour integral is very suitable for large-scale power systems due to the element distribution of descriptor matrices makes it possible to construct only one fixed matrix at all integral points. Moreover, the calculation efficiency is further improved by calculating implicitly without destroying the sparsity of descriptor systems. Numerical experiments indicate that the proposed method can significantly improve accuracy without considerably increasing the calculation time. And the approximate eigenvalue number can fully meet the needs of subsequent algorithms based on contour integral to calculate the specific eigenvalues.

Analytical solutions of a class of matrix function optimization problems with unitary constraints

In this paper we extend Theorems 3.1 and 3.2 of Xu et al. (2020) to more general cases and propose analytic solutions of the following constrained matrix maximization problems with unitary constraints: maxSkSkH=In,LkLkH=ImdetcIm+∏k=1sAkSkBkLkandmaxSkSkH=In,LkLkH=ImtrcIm+∏k=1sAkSkBkLk, where c is a complex number, Im denotes the m-order identity matrix, Ak,Bk are m×n and n×m complex matrices, and det(⋅),tr(⋅) denote the matrix determinant and trace functions. This is a non-convex nonlinear constrained matrix maximization problem. The new results improve the corresponding existing ones in Xu et al. (2020).

Biquandle brackets and knotoids

Biquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this paper, we use biquandle brackets to enhance the biquandle counting matrix invariant defined by the first two authors in (N. Gügümcü and S. Nelson, Biquandle coloring invariants of knotoids, J. Knot Theory Ramif. 28(4) (2019) 1950029). We provide examples to illustrate the method of calculation and to show that the new invariants are stronger than the previous ones. As an application we show that the trace of the biquandle bracket matrix is an invariant of the virtual closure of a knotoid.

On the optimality of the Kitanidis filter for state estimation rejecting unknown inputs

As a natural extension of the Kalman filter to systems subject to arbitrary unknown inputs, the Kitanidis filter has been designed by one-step minimization of the trace of the state estimation error covariance matrix. In this technical communiqué, it is shown that the Kitanidis filter is also optimal for the whole gain sequence in the sense of matrix positive definiteness, which notably implies that the Kitanidis filter minimizes not only the trace criterion, but also the matrix spectral norm criterion.

On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants

Randomized trace estimation is a popular and well-studied technique that approximates the trace of a large-scale matrix B by computing the average of x^T Bx for many samples of a random vector X. Often, B is symmetric positive definite (SPD) but a number of applications give rise to indefinite B. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite B with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite B, reducing the number of required samples by a factor n or even more, where n is the size of B. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.

Remdesivir Strongly Binds to RNA-Dependent RNA Polymerase, Membrane Protein, and Main Protease of SARS-CoV-2: Indication From Molecular Modeling and Simulations.

Development of new drugs is a time-taking and expensive process. Comprehensive efforts are being made globally toward the search of therapeutics against SARS-CoV-2. Several drugs such as remdesivir, favipiravir, ritonavir, and lopinavir have been included in the treatment regimen and shown effective results in several cases. Among the existing broad-spectrum antiviral drugs, remdesivir is found to be more effective against SARS-CoV-2. Remdesivir has broad-spectrum antiviral action against many single-stranded RNA viruses including pathogenic SARS-CoV and Middle East respiratory syndrome coronavirus (MERS-CoV). In this study, we proposed that remdesivir strongly binds to membrane protein (Mprotein), RNA-dependent RNA polymerase (RDRP), and main protease (Mprotease) of SARS-CoV-2. It might show antiviral activity by inhibiting more than one target. It has been found that remdesivir binds to Mprotease, Mprotein, and RDRP with −7.8, −7.4, and −7.1 kcal/mol, respectively. The structure dynamics study suggested that binding of remdesivir leads to unfolding of RDRP. It has been found that strong binding of remdesivir to Mprotein leads to decrease in structural deviations and gyrations. Additionally, the average solvent-accessible surface area of Mprotein decreases from 127.17 to 112.12 nm2, respectively. Furthermore, the eigenvalues and the trace of the covariance matrix were found to be low in case of Mprotease–remdesivir, Mprotein–remdesivir, and RDRP–remdesivir. Binding of remdesivir to Mprotease, Mprotein, and RDRP reduces the average motions in protein due to its strong binding. The MMPBSA calculations also suggested that remdesivir has strong binding affinity with Mprotein, Mprotease, and RDRP. The detailed analysis suggested that remdesivir has more than one target of SARS-CoV-2.

First-Principles Calculation of the Optical Rotatory Power of Periodic Systems: Application on α-Quartz, Tartaric Acid Crystal, and Chiral (n,m)-Carbon Nanotubes.

The self-consistent coupled-perturbed (SC-CP) method in the CRYSTAL program has been adapted to obtain electromagnetic optical rotation properties of chiral periodic systems based on the calculation of the magnetic moment induced by the electric field. Toward that end, an expression for the magnetic transition moment is developed, which involves an appropriate electronic angular momentum operator. This operator is forced to be hermitian so that the chiroptical properties are real. In our formulation, the trace of the optical rotatory power matrix is gauge-origin-invariant as long as the electric dipole transition matrix elements are obtained using the velocity (rather than position) operator. On the other hand, the component along the optic axis is invariant in general for uniaxial and biaxial crystals. Under the same conditions, these properties also do not depend on the so-called missing integers that occur in the treatment of the electric dipole moment of quasi-one-dimensional periodic systems or the analogue of missing integers for the case of higher dimensionality. Tests on a model H2O2 polymer confirm the formalism and, as desired, show that the calculated properties are independent of the size and definition of the unit cell. In addition, an empirical relation to a finite oligomer gauge-including atomic orbital (GIAO) calculation is found. Applications, with comparison to experiment, are carried for α-quartz, tartaric acid crystal, and carbon nanotubes. Future developments of this initial approach to chiroptical properties in the solid state are noted.

Resilience in Multirobot Multitarget Tracking With Unknown Number of Targets Through Reconfiguration

We address the problem of maintaining resource availability in a networked multirobot team performing distributed tracking of an unknown number of targets in a bounded environment. Robots are equipped with sensing and computational resources, enabling them to cooperatively track a set of targets using a distributed probability hypothesis density (PHD) filter. We use the trace of a robot's sensor measurement noise covariance matrix to quantify its sensing quality. While executing the tracking task, if a robot experiences sensor quality degradation, the team's communication network is reconfigured such that the robot with the faulty sensor may share information with other robots to improve the team's target-tracking ability without enforcing a large change in the number of active communication links. A central system monitor executes the network reconfiguration computations. We consider two different PHD fusion methods and propose four different mixed-integer semi-definite programming (MISDP) formulations (two formulations for each PHD fusion method) to accomplish our objective. All MISDP formulations are validated in simulation.

Trace of Positive Integer Power of Squared Special Matrix

The rectangle matrix to be discussed in this research is a special matrix where each entry in each line has the same value which is notated by An. The main aim of this paper is to find the general form of the matrix trace An powered positive integer m. To prove whether the general form of the matrix trace of An powered positive integer can be confirmed, mathematics induction and direct proof are used.

Signals classification based on IA-optimal CNN.

The versatility of the existing A-optimal-based CNN for solving multiple types of signals classification problems has not been verified by different signals datasets. Moreover, the existing A-optimal-based CNN uses a simplified approximate function as the optimization objective function instead of precise analytical function, which affects the signals classification accuracy to a certain extent. In this paper, a classification method called IA-optimal CNN is proposed. To improve the stability of the classifier, the trace of the covariance matrix of the weights of the fully connected layer is used as the optimization objective function, and the parameter optimization model is established without any simplification of the optimization objective function. In addition, to avoid the difficulty of not being able to obtain the analytical expression formula of the partial derivative of the inverse matrix with regard to the networks parameters, a novel dual function is introduced to transform the optimization problem into an equivalent binary function optimization problem. Furthermore, based on the above analytical solution results, the parameters are updated using the alternate iterative optimization method and the accurate weight update formula is deduced in detail. Five signals datasets are used to test the universality of the IA-optimal CNN in signals classification fields. The performance of IA-optimal CNN is showed, and the experimental results are compared with the existing A-optimal-based classification algorithm. Lastly, the following conclusion is proved theoretically: For the A-optimal-based CNN, the trace of the covariance matrix will continue to decrease and approach a convergence value in the iterative process, but it is impossible for the networks to strictly reach the A-optimal state.

Tree Search Techniques for Adversarial Target Tracking With Distance-Dependent Measurement Noise

We study the problem of devising a closed-loop strategy to control the position of a robot that is tracking a possibly moving target. The robot obtains noisy measurements of the target's position. The measurement noise depends on the relative states of the robot and the target. We consider scenarios where the measurement values are chosen by an adversary, so as to maximize the estimation error. Furthermore, the target may be actively evading the robot. Our main contribution is to devise a closed-loop control policy for distance-dependent target tracking that plans for a sequence of control actions, instead of acting greedily. We consider a game-theoretic formulation of the problem and seek to minimize the maximum uncertainty (trace of the posterior covariance matrix) over all possible measurement values. We exploit the structural properties of a Kalman Filter to build a policy tree that is orders of magnitude smaller than naive enumeration while still preserving optimality guarantees. We show how to obtain even more computational savings by relaxing the optimality guarantees. The resulting algorithms are evaluated through simulations and experiments with real robots.

THOR, Trace-based Hardware-driven Layer-Oriented Natural Gradient Descent Computation

It is well-known that second-order optimizer can accelerate the training of deep neural networks, however, the huge computation cost of second-order optimization makes it impractical to apply in real practice. In order to reduce the cost, many methods have been proposed to approximate a second-order matrix. Inspired by KFAC, we propose a novel Trace-based Hardware-driven layer-ORiented Natural Gradient Descent Computation method, called THOR, to make the second-order optimization applicable in the real application models. Specifically, we gradually increase the update interval and use the matrix trace to determine which blocks of Fisher Information Matrix (FIM) need to be updated. Moreover, by resorting the power of hardware, we have designed a Hardware-driven approximation method for computing FIM to achieve better performance. To demonstrate the effectiveness of THOR, we have conducted extensive experiments. The results show that training ResNet-50 on ImageNet with THOR only takes 66.7 minutes to achieve a top-1 accuracy of 75.9 % under an 8 Ascend 910 environment with MindSpore, a new deep learning computing framework. Moreover, with more computational resources, THOR can only takes 2.7 minutes to 75.9 % with 256 Ascend 910.

Expanding the Range of Hierarchical Equations of Motion by Tensor-Train Implementation.

The non-equilibrium thermo-field dynamics formulation of the hierarchical equations of motion combined with the tensor-train representation of the density matrix is discussed, and a new numerical integration scheme is introduced. The numerical methodology is based on an adaptive low-rank Galerkin reduction scheme and can preserve linear invariants (such as the trace of the density matrix). The method is applied to the study of the charge transfer dynamics in model pentacene molecular aggregates. The combined effect of a discrete set of molecular vibrational modes and a thermal bath is investigated, paying special attention to the coherent-incoherent transition of the charge transport. The new computational framework is shown to be a very promising methodology for the study of the quantum dynamics of complex molecular systems in the condensed phase.

Layer potential theory for the anisotropic Stokes system with variable L∞ symmetrically elliptic tensor coefficient

The aim of this paper is to develop a layer potential theory in L2‐based weighted Sobolev spaces on Lipschitz bounded and exterior domains of , n ≥ 3, for the anisotropic Stokes system with L∞ viscosity tensor coefficient satisfying an ellipticity condition for symmetric matrices with zero matrix trace. To do this, we explore equivalent mixed variational formulations and prove the well‐posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of , with the given data in L2‐based weighted Sobolev spaces. These results are used to define the volume (Newtonian) and layer potentials and to obtain their properties. Then, we analyze the well‐posedness of the exterior Dirichlet and Neumann problems for the anisotropic Stokes system with L∞ symmetrically elliptic tensor coefficient by representing their solutions in terms of the obtained volume and layer potentials.