##### Author

Tasissa, Abiy

##### Other Contributors

Lai, Rongjie; Kramer, Peter Roland, 1971-; Wang, Meng; Xu, Yangyang;

##### Date Issued

2019-08

##### Subject

Mathematics

##### Degree

PhD;

##### Terms of Use

This electronic version is a licensed copy owned by Rensselaer Polytechnic Institute, Troy, NY. Copyright of original work retained by author.;

##### Abstract

The last part of the thesis studies a convex quadratic program for the graph matching problem. First, we consider a simple; The technique of convex relaxation is an ubiquitous tool to solve exactly or approximately otherwise intractable nonconvex problems that appear in solving many applied problems. The technique naturally results a convex optimization program for which established convex optimization theory can be readily applied. In addition, the program is computationally tractable since it can be solved in polynomial time leveraging existing convex solvers. In recent years, convex relaxations methods such as l1 minimization and nuclear norm minimization have been successfully applied to central problems in compressive sensing and matrix completion. The main theme of this thesis is to propose and study convex relaxations in new applications and extend existing theoretical analysis for the matrix completion problem.; The matrix completion problem is a problem of recovering a low rank matrix given few entries. The second part of this thesis extends the theoretical analysis of the matrix completion problem. We study the problem of recovering a low rank matrix given a few measurements with respect to any basis. Our analysis is based on dual certificates and dual basis approach but does not assume the restricted isometry property (RIP) condition. We propose a novel sufficient condition for the analysis named the correlation condition. For matrix completion problems with respect to a structured deterministic basis, RIP might not hold or is NP hard to verify. However, the correlation condition can be checked in time O(n^3). Under the assumption that the correlation holds and the true low rank matrix obeys certain coherence conditions, the main result shows that the true matrix can be recovered with very high probability from O(nr log^2n) uniformly random expansion coefficients.; regularization which ensures that the optimal solution of the proposed program has a unique global optimum. We also propose a regularization via a linear assignment term using a cost matrix that depends on the adjacency matrices. Computationally, we develop an algorithm for the graph matching problem based on the projected gradient descent scheme. Numerical tests on challenging graphs shows that the proposed algorithm performs well in recovering the true permutation map.; The first part of this thesis studies the Euclidean distance geometry (EDG) problem, the problem of recovering the coordinates of n points given few inter-point distance measurements. The problem can be formulated as a matrix completion problem of recovering a low rank r Gram matrix with respect to a certain predefined basis. The well known restricted isometry property can not be applied to this formulation. We propose a dual basis approach for the theoretical analysis of the proposed program. Under the assumption that the Gram matrix satisfies a certain coherence condition, the main result shows that the true configuration of n points can be recovered with high probability from O(nr log^2 n ) uniformly random samples of the distance matrix. We develop a fast algorithm to solve the EDG problem. Numerical experiments on different three dimensional data and protein molecules show that the algorithm is accurate and efficient.;

##### Description

August 2019; School of Science

##### Department

Dept. of Mathematical Sciences;

##### Publisher

Rensselaer Polytechnic Institute, Troy, NY

##### Relationships

Rensselaer Theses and Dissertations Online Collection;

##### Access

Restricted to current Rensselaer faculty, staff and students. Access inquiries may be directed to the Rensselaer Libraries.;