Solving Problems with Projections

It is a curious fact that even notoriously difficult computational problems can be expressed in the form of a high-dimensional Venn diagram, where solutions lie in the overlap of a pair of remarkably simple sets, A and B. The simplicity of these sets enables operations called projections that locate the nearest point of A, or B, starting anywhere within the high-dimensional space. 
This book introduces a novel method for tackling complex problems that exploits projections and the two-set structure, offering an effective alternative to traditional, gradient-based approaches. Beginning with phase retrieval, where A and B address the properties of an image and its Fourier transform, it progresses to more diverse challenges, such as sphere packing, origami design, sudoku and tiling puzzles, data dimension reduction, and neural network training. The text presents a detailed description of this powerful and original approach and is essential reading for physicists and applied mathematicians.

• Developed over more than two decades, this book introduces several innovative algorithms that challenge the dominance of the gradient method in most large-scale computational projects
• Approximately half of the book focuses on the 'bipartisan' formulation of problems-an alternative to traditional optimization and cost/loss functions. These bipartisan approaches are inherently more creative and can uncover novel structures within problems
• Written in an engaging and accessible style, this book is an ideal resource for graduate students, scientists, engineers, and applied mathematicians working on phase retrieval problems and related fields of research