Tradeoffs Between Convergence Speed and Reconstruction Accuracy in Inverse Problems

Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is willing to tolerate, an important question is whether it is po...

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Veröffentlicht in:IEEE transactions on signal processing Jg. 66; H. 7; S. 1676 - 1690
Hauptverfasser: Giryes, Raja, Eldar, Yonina C., Bronstein, Alex M., Sapiro, Guillermo
Format: Journal Article
Sprache:Englisch
Veröffentlicht: IEEE 01.04.2018
Schlagworte:
Acceleration
approximate computing
approximation methods
Compressed sensing
Convergence
convergence of numerical methods
gradient methods
Iterative algorithms
iterative methods
neural networks
Noise measurement
optimization
Signal processing algorithms
Training
ISSN:1053-587X, 1941-0476
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Zusammenfassung:Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is willing to tolerate, an important question is whether it is possible to modify the original iterations to obtain faster convergence to a minimizer achieving the allowed error without increasing the computational cost of each iteration considerably. Relying on recent recovery techniques developed for settings in which the desired signal belongs to some low-dimensional set, we show that using a coarse estimate of this set may lead to faster convergence at the cost of an additional reconstruction error related to the accuracy of the set approximation. Our theory ties to recent advances in sparse recovery, compressed sensing, and deep learning. Particularly, it may provide a possible explanation to the successful approximation of the <inline-formula><tex-math notation="LaTeX">\ell _1</tex-math> </inline-formula>-minimization solution by neural networks with layers representing iterations, as practiced in the learned iterative shrinkage-thresholding algorithm.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2018.2791945