Sparse Signal Recovery via Generalized Entropy Functions Minimization

In this letter, the compressive sensing (CS) is utilized to accelerate the matrix filling procedure of the traditional method of moments (MoM). Instead of constituting a fully populated matrix as the traditional MoM typically does, the CS technology allows one to establish an underdetermined linear system, and the unknown induced currents can be reconstructed by finding the smallest possible $\ell _1 $ norm solutions to the underdetermined linear system. Thus, the number of computations required for constructing the impedance matrix and for solving the unknown induced currents can be reduced dramatically. The ideas can be exploited in the context of bistatic scattering from an arbitrary target. Numerical simulations are presented, and the results are compared to those obtained from traditional MoM to illustrate the validity and the efficiency of the proposed method.

In this paper, two novel algorithms are developed to design sparse linear-phase (LP) FIR filters. Compared to traditional design methods, they can jointly optimize coefficient sparsity and order of an LP FIR filter, so as to achieve a balance between filtering performance and implementation efficiency. The design problem under consideration is formally cast as a regularized l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm minimization problem, which is then tackled by two different design algorithms. In the first proposed algorithm, the objective function of the original design problem is replaced by its upper bound, which leads to a weighted l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm minimization problem, while in the second one a group of auxiliary variables are introduced such that the original design problem can be equivalently transformed to another weighted l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm minimization problem. The iterative-reweighted-least-squares (IRLS) algorithm is employed with appropriate modifications to solve both weighted l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm minimization problems. Simulation results show that, compared to traditional approaches, the proposed algorithms can achieve comparable or better design results in terms of both sparsity and effective filter order.

Optimization problems are used to model many real-life problems. Therefore, solving these problems is one of the most important goals of algorithm design. A general optimization problem can be defined by specifying a set of constraints that defines a subset in some underlying space (like the Euclidean space) called the feasible subset and an objective function that we are trying to maximize or minimize, as the case may be, over the feasible set. The difficulty of solving such problems typically depends on how ‘complex’ the feasible set and the objective function are. For example, a very important class of optimization problems is linear programming. Here the feasible subset is specified by a set of linear inequalities (in the Euclidean space); the objective function is also linear. A more general class of optimization problems is convex programming, where the feasible set is a convex subset of a Euclidean space and the objective function is also convex. Convex programs (and hence, linear programs) have a nice property that any local optimum is also a global optimum for the objective function. There are a variety of techniques for solving such problems – all of them try to approach a local optimum (which we know would be a global optimum as well). These notions are discussed in greater detail in a later section in this chapter. The more general problem, the so-called non-convex programs, where the objective function and the feasible subset could be arbitrary can be very challenging to solve. In particular, discrete optimization problems, where the feasible subset could be a (large) discrete subset of points falls under this category.

This paper presents an effective adaptive beamforming method based on an $\ell _{2}$ -norm minimization problem with multiple constraints, named as MC- $\ell _{2}$ -M algorithm. In the proposed method, the output power minimization problem can be formulated as an $\ell _{2}$ -norm minimization problem of the weight vector, with multiple constraints including distortionless response constraint and zero-forcing response constraints toward the desired signal and the interference signals, respectively. Furthermore, via an interference subspace estimation, the presented $\ell _{2}$ -norm minimization problem can be rewritten as an equivalent constrained convex optimization problem of the weight vector, which is readily to be solved. Finally, the closed-form solution of the proposed optimization problem is given. Compared with the previous works, the proposed MC- $\ell _{2}$ -M approach provides better beamforming performance, for example: 1) it reduces the sensitivity to the variation of signal-to-noise ratio (SNR) so that it has better output array gain, even if in the large dynamic SNR changing circumstances; 2) the better sidelobe and nulling levels are maintained, since it weakens the component of the desired signal in the covariance matrix of the array snapshot vector by a new defined blocking matrix; and 3) it can not only effectively suppress the interference signals but also ensure that the response of the desired signal is distortionless. Simulation results demonstrate the efficiency of the presented approach.

In communications networks, the capacity region of multisource network coding is given in terms of the set of entropy functions Gamma*. More broadly, determination of Gamma* would have an impact on converse theorems for multi-terminal problems in information theory. This paper provides several new dualities between entropy functions and network codes. Given a function g ges 0 defined on all subsets of N random variables, we provide a construction for a network multicast problem which is ldquosolvablerdquo if and only if g is the entropy function of a set of quasi-uniform random variables. The underlying network topology is fixed and the multicast problem depends on g only through link capacities and source rates. A corresponding duality is developed for linear network codes, where the constructed multicast problem is linearly solvable if and only if g is linear group characterizable. Relaxing the requirement that the domain of g be subsets of random variables, we obtain a similar duality between polymatroids and the linear programming bound. These duality results provide an alternative proof of the insufficiency of linear (and abelian) network codes, and demonstrate the utility of non-Shannon inequalities to tighten outer bounds on network coding capacity regions.

Sparse signal priors help in a variety of modern signal processing tasks. In many cases, a sparse signal needs to be recovered from an underdetermined system of equations. For instance, sparse approximation of a signal with an overcomplete dictionary or reconstruction of a sparse signal from a small number of linear measurements. The reconstruction problem typically requires solving an ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm minimization problem. In this paper we present homotopy based algorithms to update the solution of some ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> problems when the system is updated by adding new rows or columns to the underlying system matrix. We also discuss a case where these ideas can be extended to accommodate for more general changes in the system matrix.

Random-walk theory suggests that stock prices are random fluctuations over time, so the stock price is a random variable over time. Base on the theory of Brownian motion, the random-walk theory, and the maximum entropy principle. We can build a forecasting model of the continuous maximum entropy stock price. The objective function is a maximum entropy function. In the maximum entropy function, the stock price as a continuous random variable. Because in this paper we make the present value of the stock at time t as the consumer's utility function, so we can make a prior condition as the constraint conditions. The utility function will be content the prior condition. In order to make it easily for us to obtain the probability distribution, we can make the stock price as the random variable into a new random variable of contains the logarithm. So the model is transformed into a new model, named continuous maximum entropy stock price forecasting model. Because the optimization problem is convex programming, so we can construct the Lagrange function to solve the continuous maximum entropy stock price forecasting model. To the end, when the density function submits to the normal distribution, the entropy function will be the maximum. So we can deduce the expected value of the stock price in the future. As in the real stock trading will generate some action costs, the paper adds he transaction costs into the utility function, there will be make the model more realistic, which can more accurately predict future stock market stock prices.

Impulsive blind deconvolution (IBD) is a popular method to recover impulsive sources for bearing fault diagnosis. Its underpinnings are in the design of objective functions based on prior knowledge of impulsive sources and a transfer function to describe transmission path influences. However, popular objective functions cannot retain waveform impulsiveness and periodicity cyclostationarity simultaneously, and the single convolution operation of IBD methods is insufficient to describe transmission paths composed of multiple linear and nonlinear units. Inspired by the MaxPooling period modulation intensity (MPMI) and convolutional sparse learning (CSL), an adaptive multi-D-norm-driven sparse unfolding deconvolution network (AMD-SUDN) is proposed in this paper. The core strategy is that one target vector with simultaneous impulsiveness and cyclostationarity is constructed automatically through the MPMI; then, this vector is substituted into the multi D-norm to design objective functions. Moreover, an iterative soft threshold algorithm (ISTA) for the CSL model is derived, and its iterative steps are unfolded into one deconvolution network. The algorithm's performance and the hyperparameter configuration are investigated by a set of numerical simulations. Finally, the proposed AMD-SUDN is applied to detect the impulsive features of bearing faults. All comparative results verify that the proposed AMD-SUDN achieves a better deconvolution accuracy than state-of-the-art IBD methods.

The spatial patterns and processes of cities can be described with various entropy functions. However, spatial entropy always depends on the scale of measurement, and it is difficult to find a characteristic value for it. In contrast, fractal parameters can be employed to characterize scale-free phenomena and reflect the local features of random multi-scaling structure. This paper is devoted to exploring the similarities and differences between spatial entropy and fractal dimension in urban description. Drawing an analogy between cities and growing fractals, we illustrate the definitions of fractal dimension based on different entropy concepts. Three representative fractal dimensions in the multifractal dimension set, capacity dimension, information dimension, and correlation dimension, are utilized to make empirical analyses of the urban form of two Chinese cities, Beijing and Hangzhou. The results show that the entropy values vary with the measurement scale, but the fractal dimension value is stable is method and study area are fixed; if the linear size of boxes is small enough (e.g., &lt;1/25), the linear correlation between entropy and fractal dimension is significant (based on the confidence level of 99%). Further empirical analysis indicates that fractal dimension is close to the characteristic values of spatial entropy. This suggests that the physical meaning of fractal dimension can be interpreted by the ideas from entropy and scaling and the conclusion is revealing for future spatial analysis of cities.

This study aims at the problem of reconstruction the unknown links in debt networks among enterprises. We use the topological matrix of the enterprise debt network as the object of reconstruction and use the time series data of accounts receivable and payable as input and output information in the debt network to establish an underdetermined linear system about the topological matrix of the debt network. We establish an iteratively reweighted least-squares algorithm, which is an algorithm in compressed sensing. This algorithm uses reweighted ell _2-minimization to approximate ell _1-norm of the target vectors. We solve the ell _1-minimization problem of the underdetermined linear system using the iteratively reweighted least-squares algorithm and obtain the reconstructed topological matrix of the debt network. Simulation experiments show that the topology matrix reconstruction method of enterprise debt networks based on compressed sensing can reconstruct over 70% of the unknown network links, and the error is controlled within 2%.

MEMPSODE: A global optimization software based on hybridization of population-based algorithms and local searches

In this paper, we propose an efficient algorithm for finding the minimum-norm point in the intersection of a polytope and an affine set in an n-dimensional Euclidean space, where the polytope is expressed as the convex hull of finitely many points and the affine set is expressed as the intersection of k hyperplanes, k≥1. Our algorithm solves the problem by using directly the original points and the hyperplanes, rather than treating the problem as a special case of the general quadratic programming problem. One of the advantages of our approach is that our algorithm works as well for a class of problems with a large number (possibly exponential or factorial in n) of given points if every linear optimization problem over the convex hull of the given points is solved efficiently. The problem considered here is highly degenerate, and we take care of the degeneracy by solving a subproblem that is a conical version of the minimum-norm point problem, where points are replaced by rays. When the number k of hyperplanes expressing the affine set is equal to one, we can easily avoid degeneracy, but this is not the case for k≥2. We give a subprocedure for treating the degenerate case. The subprocedure is interesting in its own right. We also show the practical efficiency of our algorithm by computational experiments.

Solution of optimal reactive power dispatch of power systems using hybrid particle swarm optimization and imperialist competitive algorithms

Exploiting rank deficiency and transform domain sparsity for MR image reconstruction

The real-world decision making often involves a comparison of uncertain systems or alternatives based on fuzzy evaluations. The concept of fuzzy entropy is quite useful in such situations. However, fuzzy entropy and the conventional probabilistic entropy differ in their semantics. This article critically examines the existing fuzzy entropy functions and redefine them to bring them closer to the probabilistic entropy. More specifically, new variants of the extant Luca and Termini, and Pal and Pal fuzzy entropy functions are proposed. The proposed fuzzy entropy functions are extended for the probabilistic-fuzzy uncertainty, commonly observed in the real world. A real application is included to demonstrate the usefulness of the proposed entropy functions in decision making applications.