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ISBN 10: 1351624145
ISBN 13: 9781351624145
Author: Yun Bin Zhao
Seeking sparse solutions of underdetermined linear systems is required in many areas of engineering and science such as signal and image processing. The efficient sparse representation becomes central in various big or high-dimensional data processing, yielding fruitful theoretical and realistic results in these fields. The mathematical optimization plays a fundamentally important role in the development of these results and acts as the mainstream numerical algorithms for the sparsity-seeking problems arising from big-data processing, compressed sensing, statistical learning, computer vision, and so on. This has attracted the interest of many researchers at the interface of engineering, mathematics and computer science. Sparse Optimization Theory and Methods presents the state of the art in theory and algorithms for signal recovery under the sparsity assumption. The up-to-date uniqueness conditions for the sparsest solution of underdertemined linear systems are described. The results for sparse signal recovery under the matrix property called range space property (RSP) are introduced, which is a deep and mild condition for the sparse signal to be recovered by convex optimization methods. This framework is generalized to 1-bit compressed sensing, leading to a novel sign recovery theory in this area. Two efficient sparsity-seeking algorithms, reweighted l1-minimization in primal space and the algorithm based on complementary slackness property, are presented. The theoretical efficiency of these algorithms is rigorously analysed in this book. Under the RSP assumption, the author also provides a novel and unified stability analysis for several popular optimization methods for sparse signal recovery, including l1-mininization, Dantzig selector and LASSO. This book incorporates recent development and the author’s latest research in the field that have not appeared in other books.
Sparse Optimization Theory and Methods 1st Table of contents:
1 Uniqueness of the Sparsest Solution of Linear Systems
1.1 Introduction
1.2 Spark
1.3 Uniqueness via Mutual Coherence
1.4 Improved Uniqueness Criteria via Coherence Rank
1.4.1 Sub-Mutual Coherence and Coherence Rank
1.4.2 Improved Lower Bounds of Spark(A)
1.4.3 Improved Coherence Conditions
1.5 Babel Function and Sub-Babel Function
1.6 Notes
2 Uniqueness of Solutions to ℓ1-Minimization Problems
2.1 Strict Complementary Slackness Property (SCSP)
2.2 Least ℓ1-Norm Solution
2.2.1 Preliminary
2.2.2 Necessary Condition (I): Range Space Property of AT
2.2.3 Necessary Condition (II): Full-Column-Rank Property
2.2.4 Sufficient Condition
2.2.5 Uniqueness Characterization
2.3 Least ℓ1-Norm Non-negative Solution
2.4 Least ℓ1-Norm Points in Polyhedra
2.4.1 Restricted Range Space Property of AT
2.4.2 Proof of Necessity
2.4.3 Proof of Sufficiency
2.5 Notes
3 Equivalence of ℓ0- and ℓ1-Minimization
3.1 Equivalence and Strong Equivalence
3.2 Standard ℓ0- and ℓ1-Minimization Problems
3.3 Problems with Non-negative Constraints
3.4 Application to Linear Programming
3.5 Equivalence of ℓ0-Problem and Weighted ℓ1-Problem
3.6 Sparse Vector Recovery
3.6.1 Uniform Recovery: RSP-Based Analysis
3.6.2 Beyond Uniform Recovery
3.7 Sparse Non-negative Vector Recovery
3.7.1 Uniform Recovery: RSP+-Based Analysis
3.7.2 Non-uniform Recovery of Non-negative Vectors
3.8 Notes
4 1-Bit Compressed Sensing
4.1 Introduction
4.2 Sign Measurements and Recovery Criteria
4.3 Relaxation Models
4.4 Consistency Condition
4.4.1 Nonstandard Sign Function
4.4.2 Standard Sign Function
4.5 Reformulation of 1-Bit Compressed Sensing
4.6 Non-uniform Sign Recovery
4.7 Uniform Sign Recovery
4.8 Notes
5 Stability of Linear Sparse Optimization Methods
5.1 Introduction
5.2 Hoffman’s Error Bound for Linear Systems
5.3 Weak RSP of Order k of AT
5.4 Stability of Standard ℓ1-Minimization
5.5 Linear Dantzig Selector
5.6 Special Cases
5.6.1 Standard Dantzig Selector
5.6.2 Weighted Dantzig Selector
5.6.3 ℓ1-Minimization with ℓ∞-Norm Constraints
5.6.4 ℓ1-Minimization with ℓ1-Norm Constraints
5.7 Notes
6 Stability of Nonlinear Sparse Optimization Methods
6.1 Introduction
6.2 Orthogonal Projection Operator
6.3 Polytope Approximation of Unit Balls
6.3.1 ℓ2-Ball
6.3.2 General Unit Ball
6.4 A Necessary Condition for Stability
6.5 ℓ1-Minimization with ℓ2-Norm Constraints
6.6 Nonlinear Dantzig Selector
6.7 The LASSO Problem
6.8 Summary
6.9 Notes
7 Reweighted ℓ1-Algorithms
7.1 Merit Function for Sparsity
7.1.1 ℳ(1)-Class Merit Functions
7.1.2 ℳ(2)-Class Merit Functions
7.2 Reweighted ℓ1-Methods
7.2.1 Examples of Weights
7.3 Numerical Experiments
7.4 Theoretical Analysis
7.4.1 Well-Definedness of Algorithms
7.4.2 Convergence to Sparse Points
7.5 Summary
7.6 Notes
8 Sparsity via Dual Density
8.1 Introduction
8.2 ℓ0-Minimization with Non-negative Constraints
8.2.1 Optimal Weight via Dual Density
8.2.2 Dual-Density-Based Reweighted ℓ1-Algorithm
8.2.3 Numerical Experiments
8.2.4 Theoretical Performance
8.3 DDRW for Standard ℓ0-Minimization
8.4 Sparsity Enhancement for Weighted ℓ1-Minimizers
8.5 Notes
References
Index
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Tags: Yun Bin Zhao, Sparse Optimization, Theory and Methods