Solving Problems with Projections

Veit Elser

ISBN 10: 1009475525 / ISBN 13: 9781009475525

Published by Cambridge University Press, GB, 2025

About this Item

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.

Seller Inventory # LU-9781009475525

Bibliographic Details

• Title: Solving Problems with Projections
• Publisher: Cambridge University Press, GB
• Publication Date: 2025
• Language: English
• ISBN 10: 1009475525
• ISBN 13: 9781009475525
• Binding: Hardback
• Condition: New

About this title

Synopsis

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.

About the Author

Veit Elser, a Professor of Physics at Cornell University, is an expert in phase retrieval. His first major professional achievement was recognising that the atomic positions in quasicrystals could be understood as projections of points from a six-dimensional space into three dimensions. Projections continued to play a key role in his career, notably in 2001, when he discovered that the leading algorithm for solving the phase-retrieval problem was based on a pair of competing projections. Elser's work in physics was recognized by the William L. McMillan Prize.