Solving Problems with Projections

From Phase Retrieval to Packing

Veit Elser (Cornell University, New York)

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Forthcoming
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English

Cambridge University Press
19 June 2025

Complex analysis, complex variables; Optimisation; Mathematical physics; Applied optics; Algorithms & data structures

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.

By: Veit Elser (Cornell University New York)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
ISBN: 9781009475525
ISBN 10: 1009475525
Pages: 386
Publication Date: 19 June 2025
Audience: College/higher education , Professional and scholarly , Further / Higher Education , Undergraduate
Format: Hardback
Publisher's Status: Forthcoming

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.