Sparse Solutions of Underdetermined Linear Systems

This textbook presents a special solution to underdetermined linear systems where the number of nonzero entries in the solution is very small compared to the total number of entries. This is called a sparse solution. Since underdetermined linear systems can be very different, the authors explain how to compute a sparse solution using many approaches. Sparse Solutions of Underdetermined Linear Systems and Their Applications contains 64 algorithms for finding sparse solutions of underdetermined linear systems and their applications for matrix completion, graph clustering, and phase retrieval and provides a detailed explanation of these algorithms including derivations and convergence analysis. Exercises for each chapter help readers understand the material. This textbook is appropriate for graduate students in math and applied math, computer science, statistics, data science, and engineering. Advisors and postdoctoral scholars will also find the book interesting and useful.

Ming-Jun Lai is a professor in the Department of Mathematics at the University of Georgia. He is an approximation theorist who specializes on multivariate spline approximation, construction of multivariate wavelets and tight wavelet frames, scattered data interpolation and fitting, construction of smooth curves and surfaces, various approximation methods in numerical analysis such as sparse solutions of underdetermined linear system, numerical solution of partial differential equations, matrix completion methods, and data clustering algorithms. He has published more than 100 refereed journal papers and 36 conference papers, including in six different SIAM Journals. Dr. Lai has been on the editorial board of the journal Applied and Computational Harmonic Analysis and Journal of Applied Numerical Mathematics for many years. Yang Wang is the Dean of Science and Chair Professor of mathematics at Hong Kong University of Science and Technology. His research spans both pure and applied mathematics, including applied harmonic analysis, signal processing, fractal geometry, tiling and the application of machine learning to various practical applications. He has published over 100 papers in mathematics. He is the co-Editor-in-Chief of the journal Mathematics, Computation and Geometry of Data and is on the editorial board of Applied and Computational Harmonic Analysis and Advances in Computational Math. Dr. Wang has also served as a program director in the US National Science Foundation and is a key founding member of the Big Data Institute of HKUST.