Resolving Histogram Binning Dilemmas with Binless and Binfull Algorithms

The histogram is an analysis tool in widespread use within many sciences, with high energy physics as a prime example. However, there exists an inherent bias in the choice of binning for the histogram, with different choices potentially leading to different interpretations. This paper aims to eliminate this bias using two "debinning" algorithms. Both algorithms generate an observed cumulative distribution function from the data, and use it to construct a representation of the underlying probability distribution function. The strengths and weaknesses of these two algorithms are compared and contrasted. The applicability and future prospects of these algorithms is also discussed.Comment: 19 pages, 5 figures; additional material to be found at https://debinning.hepforge.org

Semidefinite Facial Reduction for Low-Rank Euclidean Distance Matrix Completion

The main result of this thesis is the development of a theory of semidefinite facial reduction for the Euclidean distance matrix completion problem. Our key result shows a close connection between cliques in the graph of the partial Euclidean distance matrix and faces of the semidefinite cone containing the feasible set of the semidefinite relaxation. We show how using semidefinite facial reduction allows us to dramatically reduce the number of variables and constraints required to represent the semidefinite feasible set. We have used this theory to develop a highly efficient algorithm capable of solving many very large Euclidean distance matrix completion problems exactly, without the need for a semidefinite optimization solver. For problems with a low level of noise, our SNLSDPclique algorithm outperforms existing algorithms in terms of both CPU time and accuracy. Using only a laptop, problems of size up to 40,000 nodes can be solved in under a minute and problems with 100,000 nodes require only a few minutes to solve

Computational results of a semidefinite branch-and-bound algorithm for k-cluster

International audienceThis computational paper presents a method to solve k-cluster problems exactly by intersecting semidefinite and polyhedral relaxations. Our algorithm uses a generic branch-and-bound method featuring an improved semidefinite bounding procedure. Extensive numerical experiments show that this algorithm outperforms the best known methods both in time and ability to solve large instances. For the first time, numerical results are reported for k-cluster problems on unstructured graphs with 160 vertices

Numerical solution of semidefinite constrained least squares problems

In this thesis, we are concerned with computing the least squares solution of the linear matrix equation AX = B subject to the constraint that the matrix X is positive semidefinite. For symmetric X, this is the previously studied semidefinite least squares (SDLS) problem; for nonsymmetric X, we introduce the nonsymmetric semidefinite least squares (NS-SDLS) problem. An application of this second problem is the estimation of the compliance matrix at some location on a deformable object. This is an important step in the process of making interactive computer models of de-formable objects, a technique which is used in the area of medical simulation. We also introduce a third semidefinite constrained least squares problem called the linear matrix inequality least squares (LMI-LS) problem, which is a generalization of the first two problems. Sufficient conditions for the existence and uniquenss of solutions for each of these three problems is provided. These solutions are characterized as solutions of nonlinear systems of equations known as the Karush-Kuhn-Tucker (KKT) equations. It is shown that the KKT equations for each of these problems can be stated as a semidefinite linear complementarity problem (SDLCP). Interior-point methods are proposed for the numerical solution of each of these three problems. Computational experiments are conducted which indicate that predictor-corrector interior-point methods solve these semidefinite constrained least squares problems efficiently. A noticable improvement is made over the current computational methods used for solving the SDLS problem.Science, Faculty ofMathematics, Department ofGraduat