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Decoding as a linear ill-posed problem: The entropy minimization approach

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Titel: Decoding as a linear ill-posed problem: The entropy minimization approach
Autoren: Gauthier-Umaña, Valérie, Gzyl, Henryk, ter Horst, Enrique
Weitere Verfasser: Facultad de Ingeniería::TICSw: Tecnologías de Información y Construcción de Software
Verlagsinformationen: Universidad de los Andes
Facultad de Ingeniería
Departamento de Ingeniería de Sistemas
Publikationsjahr: 2025
Bestand: Universidad de los Andes Colombia: Séneca
Schlagwörter: ill-posed inverse problems, decoding as inverse problem, convex optimization, gaussian random variables, Ingeniería
Beschreibung: The problem of decoding can be thought of as consisting of solving an ill-posed, linear inverse problem with noisy data and box constraints upon the unknowns. Specificially, we aimed to solve $\bA\bx+\be=\by,$ where $\bA$ is a matrix with positive entries and $\by$ is a vector with positive entries. It is required that $\bx\in\cK$, which is specified below, and we considered two points of view about the noise term, both of which were implied as unknowns to be determined. On the one hand, the error can be thought of as a confounding error, intentionally added to the coded message. On the other hand, we may think of the error as a true additive transmission-measurement error. We solved the problem by minimizing an entropy of the Fermi-Dirac type defined on the set of all constraints of the problem. Our approach provided a consistent way to recover the message and the noise from the measurements. In an example with a generator code matrix of the Reed-Solomon type, we examined the two points of view about the noise. As our approach enabled us to recursively decrease the $\ell_1$ norm of the noise as part of the solution procedure, we saw that, if the required norm of the noise was too small, the message was not well recovered. Our work falls within the general class of near-optimal signal recovery line of work. We also studied the case with Gaussian random matrices.
Publikationsart: article in journal/newspaper
Dateibeschreibung: 14 páginas; application/pdf
Sprache: English
Relation: 4152; 4139; 10; AIMS Mathematics; 1. F. L. Bauer, Decrypted secrets: Methods and maxims on cryptography, Berlin: Springer-Verlag, 1997. 2. J. M. Borwein, A. S. Lewis, Convex analysis and nonlinear optimization, 2nd Edition, Berlin: CMS-Springer, 2006. 3. D. Burshtein, I. Goldenberg, Improved linear programming decoding and bounds on the minimum distance of LDPC codes, IEEE Inf. Theory Work., 2010. Available from: https://ieeexplore. ieee.org/document/5592887. 4. E. Candes, T. Tao, Decoding by linear programming, IEEE Tran. Inf. Theory, 51 (2005), 4203– 4215. http://dx.doi.org/10.1109/TIT.2005.858979 5. E. Candes, T. Tao, Near optimal signal recovery from random projections: Universal encoding strategies, IEEE Tran. Inf. Theory, 52 (2006), 5406–5425. http://dx.doi.org/10.1109/TIT.2006.885507 6. C. Daskalakis, G. Alexandros, A. G. Dimakis, R. M. Karp, M. J. Wainwright, Probabilistic analysis of linear programming decoding, IEEE Tran. Inf. Theory, 54 (2008), 3565–3578. http://dx.doi.org/10.1109/TIT.2008.926452 7. S. El Rouayyheb, C. N. Georghiades, Graph theoretic methods in coding theory, Classical, Semiclass. Quant. Noise, 2012, 53–62. https://doi.org/10.1007/978-1-4419-6624-7 5 8. J. Feldman, M. J. Wainwright, D. R. Karger, Using linear programming to decode binary linear codes, IEEE Tran. Inf. Theory, 51 (2005), 954–972. https://doi.org/10.1109/TIT.2004.842696 9. F. Gamboa, H. Gzyl, Linear programming with maximum entropy, Math. Comput. Modeling, 13 (1990), 49–52. 10. Y. S. Han, A new treatment of priority-first search maximum-likelihood soft-decision decoding of linear block codes, IEEE Tran. Inf. Theory, 44 (1998), 3091–3096. https://doi.org/10.1109/18.737538 11. M. Helmiling, Advances in mathematical programming-based error-correction decoding, OPUS Koblen., 2015. Available from: https://kola.opus.hbz-nrw.de/frontdoor/index/ index/year/2015/docId/948. 12. M. Helmling, S. Ruzika, A. Tanatmis, Mathematical programming decoding of binary linear codes: Theory and algorithms, IEEE Tran. Inf. Theory, 58 (2012), 4753–4769. https://doi.org/10.1109/TIT.2012.2191697 13. M. R. Islam, Linear programming decoding: The ultimate decoding technique for low density parity check codes, Radioel. Commun. Syst., 56 (2013), 57–72. https://doi.org/10.3103/S0735272713020015 14. T. Kaneko, T. Nishijima, S. Hirasawa, An improvement of soft-decision maximum-likelihood decoding algorithm using hard-decision bounded-distance decoding, IEEE Tran. Inf. Theory, 43 (1997), 1314–1319. https://doi.org/10.1109/18.605601 15. S. B. McGrayne, The theory that would not die. How Bayes’ rule cracked the enigma code, hunted down Russian submarines, & emerged triumphant from two centuries of controversy, New Haven: Yale University Press, 2011. 16. R. J. McEliece, A public-key cryptosystem based on algebraic, Coding Th., 4244 (1978), 114–116. 17. H. Mohammad, N. Taghavi, P. H. Siegel, Adaptive methods for linear programming decoding, IEEE Tran. Inf. Theory, 54 (2008), 5396–5410. https://doi.org/10.1109/TIT.2008.2006384 18. G. Xie, F. Fu, H. Li, W. Du, Y. Zhong, L. Wang, et al, A gradient-enhanced physicsinformed neural networks method for the wave equation, Eng. Anal. Bound. Ele., 166 (2024). https://doi.org/10.1016/j.enganabound.2024.105802 19. Q. Yin, X. B. Shu, Y. Guo, Z. Y. Wang, Optimal control of stochastic differential equations with random impulses and the Hamilton-Jacobi-Bellman equation, Optimal Control Appl. Methods, 45 (2024), 2113–2135. https://doi.org/10.1002/oca.3139 20. B. Zolfaghani, K. Bibak, T. Koshiba, The odyssey of entropy: Cryptography, Entropy, 24 (2022), 266–292. https://doi.org/10.3390/e24020266; https://hdl.handle.net/1992/76128; https://doi.org/10.3934/math.2025192; instname:Universidad de los Andes; reponame:Repositorio Institucional Séneca; repourl:https://repositorio.uniandes.edu.co/
DOI: 10.3934/math.2025192
Verfügbarkeit: https://hdl.handle.net/1992/76128
https://doi.org/10.3934/math.2025192
Rights: Al consultar y hacer uso de este recurso, está aceptando las condiciones de uso establecidas por los autores ; info:eu-repo/semantics/openAccess ; http://purl.org/coar/access_right/c_abf2
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  Data: Decoding as a linear ill-posed problem: The entropy minimization approach
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  Data: The problem of decoding can be thought of as consisting of solving an ill-posed, linear inverse problem with noisy data and box constraints upon the unknowns. Specificially, we aimed to solve $\bA\bx+\be=\by,$ where $\bA$ is a matrix with positive entries and $\by$ is a vector with positive entries. It is required that $\bx\in\cK$, which is specified below, and we considered two points of view about the noise term, both of which were implied as unknowns to be determined. On the one hand, the error can be thought of as a confounding error, intentionally added to the coded message. On the other hand, we may think of the error as a true additive transmission-measurement error. We solved the problem by minimizing an entropy of the Fermi-Dirac type defined on the set of all constraints of the problem. Our approach provided a consistent way to recover the message and the noise from the measurements. In an example with a generator code matrix of the Reed-Solomon type, we examined the two points of view about the noise. As our approach enabled us to recursively decrease the $\ell_1$ norm of the noise as part of the solution procedure, we saw that, if the required norm of the noise was too small, the message was not well recovered. Our work falls within the general class of near-optimal signal recovery line of work. We also studied the case with Gaussian random matrices.
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  Data: 4152; 4139; 10; AIMS Mathematics; 1. F. L. Bauer, Decrypted secrets: Methods and maxims on cryptography, Berlin: Springer-Verlag, 1997. 2. J. M. Borwein, A. S. Lewis, Convex analysis and nonlinear optimization, 2nd Edition, Berlin: CMS-Springer, 2006. 3. D. Burshtein, I. Goldenberg, Improved linear programming decoding and bounds on the minimum distance of LDPC codes, IEEE Inf. Theory Work., 2010. Available from: https://ieeexplore. ieee.org/document/5592887. 4. E. Candes, T. Tao, Decoding by linear programming, IEEE Tran. Inf. Theory, 51 (2005), 4203– 4215. http://dx.doi.org/10.1109/TIT.2005.858979 5. E. Candes, T. Tao, Near optimal signal recovery from random projections: Universal encoding strategies, IEEE Tran. Inf. Theory, 52 (2006), 5406–5425. http://dx.doi.org/10.1109/TIT.2006.885507 6. C. Daskalakis, G. Alexandros, A. G. Dimakis, R. M. Karp, M. J. Wainwright, Probabilistic analysis of linear programming decoding, IEEE Tran. Inf. Theory, 54 (2008), 3565–3578. http://dx.doi.org/10.1109/TIT.2008.926452 7. S. El Rouayyheb, C. N. Georghiades, Graph theoretic methods in coding theory, Classical, Semiclass. Quant. Noise, 2012, 53–62. https://doi.org/10.1007/978-1-4419-6624-7 5 8. J. Feldman, M. J. Wainwright, D. R. Karger, Using linear programming to decode binary linear codes, IEEE Tran. Inf. Theory, 51 (2005), 954–972. https://doi.org/10.1109/TIT.2004.842696 9. F. Gamboa, H. Gzyl, Linear programming with maximum entropy, Math. Comput. Modeling, 13 (1990), 49–52. 10. Y. S. Han, A new treatment of priority-first search maximum-likelihood soft-decision decoding of linear block codes, IEEE Tran. Inf. Theory, 44 (1998), 3091–3096. https://doi.org/10.1109/18.737538 11. M. Helmiling, Advances in mathematical programming-based error-correction decoding, OPUS Koblen., 2015. Available from: https://kola.opus.hbz-nrw.de/frontdoor/index/ index/year/2015/docId/948. 12. M. Helmling, S. Ruzika, A. Tanatmis, Mathematical programming decoding of binary linear codes: Theory and algorithms, IEEE Tran. Inf. Theory, 58 (2012), 4753–4769. https://doi.org/10.1109/TIT.2012.2191697 13. M. R. Islam, Linear programming decoding: The ultimate decoding technique for low density parity check codes, Radioel. Commun. Syst., 56 (2013), 57–72. https://doi.org/10.3103/S0735272713020015 14. T. Kaneko, T. Nishijima, S. Hirasawa, An improvement of soft-decision maximum-likelihood decoding algorithm using hard-decision bounded-distance decoding, IEEE Tran. Inf. Theory, 43 (1997), 1314–1319. https://doi.org/10.1109/18.605601 15. S. B. McGrayne, The theory that would not die. How Bayes’ rule cracked the enigma code, hunted down Russian submarines, & emerged triumphant from two centuries of controversy, New Haven: Yale University Press, 2011. 16. R. J. McEliece, A public-key cryptosystem based on algebraic, Coding Th., 4244 (1978), 114–116. 17. H. Mohammad, N. Taghavi, P. H. Siegel, Adaptive methods for linear programming decoding, IEEE Tran. Inf. Theory, 54 (2008), 5396–5410. https://doi.org/10.1109/TIT.2008.2006384 18. G. Xie, F. Fu, H. Li, W. Du, Y. Zhong, L. Wang, et al, A gradient-enhanced physicsinformed neural networks method for the wave equation, Eng. Anal. Bound. Ele., 166 (2024). https://doi.org/10.1016/j.enganabound.2024.105802 19. Q. Yin, X. B. Shu, Y. Guo, Z. Y. Wang, Optimal control of stochastic differential equations with random impulses and the Hamilton-Jacobi-Bellman equation, Optimal Control Appl. Methods, 45 (2024), 2113–2135. https://doi.org/10.1002/oca.3139 20. B. Zolfaghani, K. Bibak, T. Koshiba, The odyssey of entropy: Cryptography, Entropy, 24 (2022), 266–292. https://doi.org/10.3390/e24020266; https://hdl.handle.net/1992/76128; https://doi.org/10.3934/math.2025192; instname:Universidad de los Andes; reponame:Repositorio Institucional Séneca; repourl:https://repositorio.uniandes.edu.co/
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      – SubjectFull: decoding as inverse problem
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