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Author: Alex Lemon
Publisher:
ISBN: 9781680831375
Category : Ranking and selection (Statistics)
Languages : en
Pages : 156
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Book Description
Finding low-rank solutions of semidefinite programs is important in many applications. For example, semidefinite programs that arise as relaxations of polynomial optimization problems are exact relaxations when the semidefinite program has a rank-1 solution. Unfortunately, computing a minimum-rank solution of a semidefinite program is an NP-hard problem. In this paper we review the theory of low-rank semidefinite programming, presenting theorems that guarantee the existence of a low-rank solution, heuristics for computing low-rank solutions, and algorithms for finding low-rank approximate solutions. Then we present applications of the theory to trust-region problems and signal processing.
Author: Alex Lemon
Publisher:
ISBN: 9781680831375
Category : Ranking and selection (Statistics)
Languages : en
Pages : 156
Get Book Here
Book Description
Finding low-rank solutions of semidefinite programs is important in many applications. For example, semidefinite programs that arise as relaxations of polynomial optimization problems are exact relaxations when the semidefinite program has a rank-1 solution. Unfortunately, computing a minimum-rank solution of a semidefinite program is an NP-hard problem. In this paper we review the theory of low-rank semidefinite programming, presenting theorems that guarantee the existence of a low-rank solution, heuristics for computing low-rank solutions, and algorithms for finding low-rank approximate solutions. Then we present applications of the theory to trust-region problems and signal processing.
Author: Alex Lemon
Publisher: Now Publishers
ISBN: 9781680831368
Category : Mathematics
Languages : en
Pages : 180
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Book Description
Finding low-rank solutions of semidefinite programs is important in many applications. For example, semidefinite programs that arise as relaxations of polynomial optimization problems are exact relaxations when the semidefinite program has a rank-1 solution. Unfortunately, computing a minimum-rank solution of a semidefinite program is an NP-hard problem. This monograph reviews the theory of low-rank semidefinite programming, presenting theorems that guarantee the existence of a low-rank solution, heuristics for computing low-rank solutions, and algorithms for finding low-rank approximate solutions. It then presents applications of the theory to trust-region problems and signal processing.
Author: Changhui Choi
Publisher:
ISBN: 9780549056898
Category :
Languages : en
Pages : 117
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Book Description
Author: Tae Jung Roh
Publisher:
ISBN: 9780549130772
Category :
Languages : en
Pages : 266
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Book Description
Much of the recent work in this field has centered around optimization problems involving nonnegative polynomial constraints. The basic observation is that sum-of-squares formulations (or relaxations) of such problems can be solved by semidefinite programming. In practice, however, the semidefinite programs that result from this approach are often challenging for general-purpose solvers due to the presence of large auxiliary matrix variables. It is therefore of interest to develop specialized algorithms for semidefinite programs derived from sum-of-squares formulations.
Author:
Publisher:
ISBN: 9789056683719
Category :
Languages : en
Pages :
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Book Description
Author: Madeleine Udell
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Principal components analysis (PCA) is a well-known technique for approximating a tabular data set by a low rank matrix. This dissertation extends the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix completion, sparse and robust PCA, k-means, k-SVD, and maximum margin matrix factorization. The method handles heterogeneous data sets, and leads to coherent schemes for compressing, denoising, and imputing missing entries across all data types simultaneously. It also admits a number of interesting interpretations of the low rank factors, which allow clustering of examples or of features. We propose several parallel algorithms for fitting generalized low rank models, and describe implementations and numerical results.
Author: Zhisu Zhu
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
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.
Author: Grigoriy Blekherman
Publisher: SIAM
ISBN: 1611972280
Category : Mathematics
Languages : en
Pages : 487
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Book Description
An accessible introduction to convex algebraic geometry and semidefinite optimization. For graduate students and researchers in mathematics and computer science.
Author: P.-A. Absil
Publisher: Princeton University Press
ISBN: 1400830249
Category : Mathematics
Languages : en
Pages : 240
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Book Description
Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.
Author: Thierry Bouwmans
Publisher: CRC Press
ISBN: 1315353539
Category : Computers
Languages : en
Pages : 510
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Book Description
Handbook of Robust Low-Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing shows you how robust subspace learning and tracking by decomposition into low-rank and sparse matrices provide a suitable framework for computer vision applications. Incorporating both existing and new ideas, the book conveniently gives you one-stop access to a number of different decompositions, algorithms, implementations, and benchmarking techniques. Divided into five parts, the book begins with an overall introduction to robust principal component analysis (PCA) via decomposition into low-rank and sparse matrices. The second part addresses robust matrix factorization/completion problems while the third part focuses on robust online subspace estimation, learning, and tracking. Covering applications in image and video processing, the fourth part discusses image analysis, image denoising, motion saliency detection, video coding, key frame extraction, and hyperspectral video processing. The final part presents resources and applications in background/foreground separation for video surveillance. With contributions from leading teams around the world, this handbook provides a complete overview of the concepts, theories, algorithms, and applications related to robust low-rank and sparse matrix decompositions. It is designed for researchers, developers, and graduate students in computer vision, image and video processing, real-time architecture, machine learning, and data mining.
