Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations

This paper establishes information-theoretic limits in estimating a finite field low-rank matrix given random linear measurements of it. These linear measurements are obtained by taking inner products of the low-rank matrix with random sensing matrices. Necessary and sufficient conditions on the number of measurements required are provided. It is shown that these conditions are sharp and the minimum-rank decoder is asymptotically optimal. The reliability function of this decoder is also derived by appealing to de Caen's lower bound on the probability of a union. The sufficient condition also holds when the sensing matrices are sparse - a scenario that may be amenable to efficient decoding. More precisely, it is shown that if the n\times n-sensing matrices contain, on average, \Omega(nlog n) entries, the number of measurements required is the same as that when the sensing matrices are dense and contain entries drawn uniformly at random from the field. Analogies are drawn between the above results and rank-metric codes in the coding theory literature. In fact, we are also strongly motivated by understanding when minimum rank distance decoding of random rank-metric codes succeeds. To this end, we derive distance properties of equiprobable and sparse rank-metric codes. These distance properties provide a precise geometric interpretation of the fact that the sparse ensemble requires as few measurements as the dense one. Finally, we provide a non-exhaustive procedure to search for the unknown low-rank matrix.Comment: Accepted to the IEEE Transactions on Information Theory; Presented at IEEE International Symposium on Information Theory (ISIT) 201

Optimal Placement of Metal Foils in Ultrasonic Consolidation Process

Ultrasonic Consolidation is a combination of additive and subtractive manufacturing processes resulting in considerable material waste. This waste is a function of the geometry of the part being manufactured and of the relative placement of the layer with respect to the metal bands. Thus the waste may be minimized by careful choice of the layer angle and offset from the original position. Previous work done in this field had developed an automated algorithm which optimally places and orients the individual slices of the STL file of the artifact being manufactured. However, the problem was solved on a 2-D scale and the 3- D nature of the part was not considered for the development of the algorithm. The earlier algorithm employed approximation on the input data to minimize the computational expense. This resulted in convergence of the optimizer to suboptimal solutions. Further, as the final part is made of anisotropic material the relative angles and overlap between subsequent layers also plays an important role in the final part strength. Finally, it is noted that the build time required for the ultrasonic consolidation process is a function of the number of bands required to form each slice. Considering these limitations and opportunities, this thesis presents an algorithm which optimally orients and places the part layers with respect to aluminum bands in order to minimize the waste formed and the build time required. The algorithm has the capability of increasing the part strength by forming crisscross and brick structures using the metal foils. This research work also improves on the previous algorithm by extending the functionality of the algorithm by building in capability to handle multiple loops within the same slice and non convex slice data. Further, the research studies the choice of optimizer that needs to be employed for different types of input data

Global optimization with piecewise linear approximation

textGlobal optimization deals with the development of solution methodologies for nonlinear nonconvex optimization problems. These problems, which could arise in diverse situations ranging from optimizing hydro-power generation schedules to estimating coefficients of non-linear regression models, are difficult for traditional nonlinear solvers that iteratively search the neighborhood around a starting point. The Piecewise Linear Approximation (PLA) method that we study in this dissertation seeks to generate ‘good’ starting points, hopefully ones that lie in the basin of attraction of the globally optimal solution. In this approach, we approximate the non-linear functions in the optimization problem by piecewise linear functions defined over the vertices of a grid that partitions the domain of each nonlinear function into cells. Based on this approximation, we convert the original nonlinear program into a mixed integer program (MIP) and use the solution to this MIP as a starting point for a local nonlinear solver. In this dissertation, we validate the effectiveness of the PLA approach as a global optimization approach by applying it to a diverse set of continuous and discrete nonlinear optimization problems. Further, we develop various modeling and algorithmic strategies for enhancing the basic approach. Our computational results demonstrate that the PLA approach works well on non-convex problems and can, in some cases, provide better solutions than those provided by existing nonlinear solvers.Information, Risk, and Operations Management (IROM

Tuberculosis Disease Detection through CXR Images based on Deep Neural Network Approach

Tuberculosis (TB) is a disease that, if left untreated for an extended period of time, can ultimately be fatal. Early TB detection can be aided by using a deep learning ensemble. In previous work, ensemble classifiers were only trained on images that shared similar characteristics. It is necessary for an ensemble to produce a diverse set of errors in order for it to be useful; this can be accomplished by making use of a number of different classifiers and/or features. In light of this, a brand-new framework has been constructed in this study for the purpose of segmenting and identifying TB in human Chest X-ray. It was determined that searching traditional web databases for chest X-ray was necessary. At this point, we pass the photos that we have collected over to Swin ResUnet3 so that they may be segmented. After the segmented chest X-ray have been provided to it, the Multi-scale Attention-based Densenet with Extreme Learning Machine (MAD-ELM) model will be applied in the detection stage in order to effectively diagnose tuberculosis from human chest X-ray. This will be done in order to maximize efficiency. Because it increased the variety of errors made by the basic classifiers, the supplied variation of the approach that was proposed was able to detect tuberculosis more effectively. The proposed ensemble method produced results with an accuracy of 94.2 percent, which are comparable to those obtained by past efforts

Improved Fitness Dependent Optimizer for Solving Economic Load Dispatch Problem

Economic Load Dispatch depicts a fundamental role in the operation of power systems, as it decreases the environmental load, minimizes the operating cost, and preserves energy resources. The optimal solution to Economic Load Dispatch problems and various constraints can be obtained by evolving several evolutionary and swarm-based algorithms. The major drawback to swarm-based algorithms is premature convergence towards an optimal solution. Fitness Dependent Optimizer is a novel optimization algorithm stimulated by the decision-making and reproductive process of bee swarming. Fitness Dependent Optimizer (FDO) examines the search spaces based on the searching approach of Particle Swarm Optimization. To calculate the pace, the fitness function is utilized to generate weights that direct the search agents in the phases of exploitation and exploration. In this research, the authors have carried out Fitness Dependent Optimizer to solve the Economic Load Dispatch problem by reducing fuel cost, emission allocation, and transmission loss. Moreover, the authors have enhanced a novel variant of Fitness Dependent Optimizer, which incorporates novel population initialization techniques and dynamically employed sine maps to select the weight factor for Fitness Dependent Optimizer. The enhanced population initialization approach incorporates a quasi-random Sabol sequence to generate the initial solution in the multi-dimensional search space. A standard 24-unit system is employed for experimental evaluation with different power demands. Empirical results obtained using the enhanced variant of the Fitness Dependent Optimizer demonstrate superior performance in terms of low transmission loss, low fuel cost, and low emission allocation compared to the conventional Fitness Dependent Optimizer. The experimental study obtained 7.94E-12.Comment: 42 page