The Inverse Problem of Positive Autoconvolution

We pose the problem of approximating optimally a given nonnegative signal with the scalar autoconvolution of a nonnegative signal. The I-divergence is chosen as the optimality criterion being well suited to incorporate nonnegativity constraints. After proving the existence of an optimal approximatio...

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Veröffentlicht in:IEEE transactions on information theory Jg. 69; H. 6; S. 4081 - 4092
Hauptverfasser: Finesso, Lorenzo, Spreij, Peter
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.06.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
alternating minimization
Approximation
Approximation algorithms
Asymptotic properties
Autoconvolution
Behavioral sciences
Convergence
Convolution
Divergence
I-divergence
inverse problem
Inverse problems
Iterative algorithms
Iterative methods
Minimization
Optimality criteria
Optimization
positive system
Signal processing algorithms
ISSN:0018-9448, 1557-9654
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Zusammenfassung:We pose the problem of approximating optimally a given nonnegative signal with the scalar autoconvolution of a nonnegative signal. The I-divergence is chosen as the optimality criterion being well suited to incorporate nonnegativity constraints. After proving the existence of an optimal approximation, we derive an iterative descent algorithm of the alternating minimization type to find a minimizer. The algorithm is based on the lifting technique developed by Csiszár and Tusnádi and exploits the optimality properties of the related minimization problems in the larger space. We study the asymptotic behavior of the iterative algorithm and prove, among other results, that its limit points are Kuhn-Tucker points of the original minimization problem. Numerical experiments confirm the asymptotic results and exhibit the fast convergence of the proposed algorithm.
Bibliographie:ObjectType-Article-1
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2023.3244407