Annals of Mathematical Sciences and Applications

Volume 3 (2018)

Number 1

Special issue in honor of Professor David Mumford, dedicated to the memory of Jennifer Mumford

Guest Editors: Stuart Geman, David Gu, Stanley Osher, Chi-Wang Shu, Yang Wang, and Shing-Tung Yau

Data recovery on a manifold from linear samples: theory and computation

Pages: 337 – 365

DOI: https://dx.doi.org/10.4310/AMSA.2018.v3.n1.a11

Authors

Jian-Feng Cai (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong)

Yi Rong (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong)

Yang Wang (Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong)

Zhiqiang Xu (LSEC, Inst. Comp. Math., Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China)

Abstract

Data recovery on a manifold is an important problem in many applications. Many such problems, e.g. compressive sensing, involve solving a system of linear equations knowing that the unknowns lie on a known manifold. The aim of this paper is to survey theoretical results and numerical algorithms about the recovery of signals lying on a manifold from linear measurements. Particularly, we focus on the case where signals lying on an algebraic variety. We first introduce the tools from algebraic geometry which plays an important role in studying the minimal measurement number and also show its applications. We finally introduce the numerical algorithms for solving it.

Keywords

frames, phase retrieval

2010 Mathematics Subject Classification

42C15

Full Text (PDF format)

Jian-Feng Cai was supported by Hong Kong Research Grant Council grant 16300616 and 16306317.

Yang Wang was supported in part by the Hong Kong Research Grant Council grant 16306415 and 16317416.

Zhiqiang Xu was supported by NSFC grant (11422113, 91630203, 11331012) and by National Basic Research Program of China (973 Program 2015CB856000).

Received 15 September 2017

Published 27 March 2018