TY - JOUR
T1 - Sparse Recovery beyond Compressed Sensing
T2 - Separable Nonlinear Inverse Problems
AU - Bernstein, Brett
AU - Liu, Sheng
AU - Papadaniil, Chrysa
AU - Fernandez-Granda, Carlos
N1 - Funding Information:
Manuscript received May 18, 2019; revised November 10, 2019; accepted March 11, 2020. Date of publication April 1, 2020; date of current version August 18, 2020. The work of Brett Bernstein was supported in part by the MacCracken Fellowship and in part by the Isaac Barkey and Ernesto Yhap Fellowship. The work of Chrysa Papadaniil was supported by NIH under Grant NEI-R01-EY025673. The work of Carlos Fernandez-Granda was supported by NSF under Grant DMS-1616340. This article was presented in part at the 2018 SIAM Annual Meeting and in part at the 2019 Asilomar Conference on Signals, Systems, and Computers. (Corresponding author: Brett Bernstein.) Brett Bernstein is with the Courant Institute of Mathematical Sciences, New York University, New York, NY 10011 USA (e-mail: brett.bernstein@ nyu.edu).
Publisher Copyright:
© 1963-2012 IEEE.
PY - 2020/9
Y1 - 2020/9
N2 - Extracting information from nonlinear measurements is a fundamental challenge in data analysis. In this work, we consider separable inverse problems, where the data are modeled as a linear combination of functions that depend nonlinearly on certain parameters of interest. These parameters may represent neuronal activity in a human brain, frequencies of electromagnetic waves, fluorescent probes in a cell, or magnetic relaxation times of biological tissues. Separable nonlinear inverse problems can be reformulated as underdetermined sparse-recovery problems, and solved using convex programming. This approach has had empirical success in a variety of domains, from geophysics to medical imaging, but lacks a theoretical justification. In particular, compressed-sensing theory does not apply, because the measurement operators are deterministic and violate incoherence conditions such as the restricted-isometry property. Our main contribution is a theory for sparse recovery adapted to deterministic settings. We show that convex programming succeeds in recovering the parameters of interest, as long as their values are sufficiently distinct with respect to the correlation structure of the measurement operator. The theoretical results are illustrated through numerical experiments for two applications: heat-source localization and estimation of brain activity from electroencephalography data.
AB - Extracting information from nonlinear measurements is a fundamental challenge in data analysis. In this work, we consider separable inverse problems, where the data are modeled as a linear combination of functions that depend nonlinearly on certain parameters of interest. These parameters may represent neuronal activity in a human brain, frequencies of electromagnetic waves, fluorescent probes in a cell, or magnetic relaxation times of biological tissues. Separable nonlinear inverse problems can be reformulated as underdetermined sparse-recovery problems, and solved using convex programming. This approach has had empirical success in a variety of domains, from geophysics to medical imaging, but lacks a theoretical justification. In particular, compressed-sensing theory does not apply, because the measurement operators are deterministic and violate incoherence conditions such as the restricted-isometry property. Our main contribution is a theory for sparse recovery adapted to deterministic settings. We show that convex programming succeeds in recovering the parameters of interest, as long as their values are sufficiently distinct with respect to the correlation structure of the measurement operator. The theoretical results are illustrated through numerical experiments for two applications: heat-source localization and estimation of brain activity from electroencephalography data.
KW - Sparse recovery
KW - convex programming
KW - correlated measurements
KW - dual certificates
KW - incoherence
KW - nonlinear inverse problems
KW - source localization
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U2 - 10.1109/TIT.2020.2985015
DO - 10.1109/TIT.2020.2985015
M3 - Article
AN - SCOPUS:85090408626
VL - 66
SP - 5904
EP - 5926
JO - IRE Professional Group on Information Theory
JF - IRE Professional Group on Information Theory
SN - 0018-9448
IS - 9
M1 - 9052719
ER -