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Sparse Optimization Theory and Methods (Paperback)
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Sparse Optimization Theory and Methods (Paperback)
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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.
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