A semidefinite programming method for graph realization and low rank matrix completion problem
- Owing to their high accuracy and ease of formulation, there has been great interest in applying convex optimization techniques, particularly semidefinite programming (SDP) relaxation, to the graph realization and sensor network localization problems in recent years. A drawback of such techniques is that the resulting convex program is often expensive to solve. In order to speed up computation, various edge sparsification heuristics have been proposed, whose aim is to reduce the number of edges in the input graph. Although these heuristics do reduce the size of the convex program and hence make it faster to solve, they are often ad hoc in nature and do not preserve the realization (or localization) properties of the input. As such, one often has to face a tradeoff between solution accuracy and computational effort. In this thesis, we propose a novel edge sparsification heuristic that can provably preserve the realization (or localization) properties of the original input. At the heart of our heuristic is a graph decomposition procedure that allows us to identify certain sparse generically universally rigid subgraphs of the input graph. Our computational results show that the proposed approach can significantly reduce the computational and memory complexities of SDP-based algorithms for solving the graph realization and sensor network localization problems. Moreover, it compares favorably with existing speedup approaches in terms of both accuracy and solution time. The graph realization problem indeed aims to reconstruct a matrix from a sampling of its entries, which can be viewed as a special case of the well-studied matrix completion problem. The main objective of the matrix completion problem is to design an efficient algorithm that can reconstruct a matrix by inspecting only a small number of its entries. Although, generally speaking, this is an impossible task, Candes and co-authors have recently shown that under a so-called incoherence assumption, a rank r n x n matrix can be reconstructed using SDP after one inspects O(nr log6 n) of its entries. We first provide an equivalent SDP formulation based on chordal decomposition, which has smaller SDP cones. Then we propose an alternative approach that can reconstruct a larger class of matrices by inspecting a significantly smaller number of the entries. Specifically, we first introduce a class of matrices, which we call stable matrices, and show that it includes all those that satisfy the incoherence assumption. Then, we propose a randomized basis pursuit (RBP) algorithm and show that it can reconstruct a stable rank r n x n matrix after inspecting O(nr log n) of its entries. Our sampling bound is only a logarithmic factor away from the information-theoretic limit and is essentially optimal.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Stanford University, Department of Computational and Mathematical Engineering.
|Saunders, Michael A
|Saunders, Michael A
|Statement of responsibility
|Submitted to the Department of Computational and Mathematical Engineering.
|Ph.D. Stanford University 2011
- © 2011 by Zhisu Zhu
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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