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4 results

  • 1 - Origins

  • Veit Elser, Cornell University, New York
  • Book:
    Solving Problems with Projections
    Published online:
    05 June 2025
    Print publication:
    19 June 2025, pp 1-22
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  • Summary

    The main ideas are introduced in a historical context. Beginning with phase retrieval and ending with neural networks, the reader will get a sense of the book’s broad scope.

Solving Problems with Projections
  • From Phase Retrieval to Packing
  • Veit Elser
  • Published online:
    05 June 2025
    Print publication:
    19 June 2025
  • 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.