Decoding as a linear ill-posed problem: The entropy minimization approach
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 $ {{\boldsymbol A}}{{\boldsymbol x}}+{{\boldsymbol e}} = {{\boldsymbol y}}, $ where $ {{\boldsymbol A}} $ is...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2025-02-01
|
| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2025192 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849703643691876352 |
|---|---|
| author | Valérie Gauthier-Umaña Henryk Gzyl Enrique ter Horst |
| author_facet | Valérie Gauthier-Umaña Henryk Gzyl Enrique ter Horst |
| author_sort | Valérie Gauthier-Umaña |
| collection | DOAJ |
| description | 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 $ {{\boldsymbol A}}{{\boldsymbol x}}+{{\boldsymbol e}} = {{\boldsymbol y}}, $ where $ {{\boldsymbol A}} $ is a matrix with positive entries and $ {{\boldsymbol y}} $ is a vector with positive entries. It is required that $ {{\boldsymbol x}}\in{{\mathcal K}} $, 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. |
| format | Article |
| id | doaj-art-4402cdce9aa549f2b347d3bfb93c9d24 |
| institution | DOAJ |
| issn | 2473-6988 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-4402cdce9aa549f2b347d3bfb93c9d242025-08-20T03:17:09ZengAIMS PressAIMS Mathematics2473-69882025-02-011024139415210.3934/math.2025192Decoding as a linear ill-posed problem: The entropy minimization approachValérie Gauthier-Umaña0Henryk Gzyl1Enrique ter Horst2Systems and Computing Engineering Department, Universidad de los Andes, Bogotá, ColombiaCenter for Finance, IESA School of Business, Caracas, VenezuelaSchool of Management, Universidad de los Andes, Bogotá, ColombiaThe 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 $ {{\boldsymbol A}}{{\boldsymbol x}}+{{\boldsymbol e}} = {{\boldsymbol y}}, $ where $ {{\boldsymbol A}} $ is a matrix with positive entries and $ {{\boldsymbol y}} $ is a vector with positive entries. It is required that $ {{\boldsymbol x}}\in{{\mathcal K}} $, 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.https://www.aimspress.com/article/doi/10.3934/math.2025192ill-posed inverse problemsdecoding as inverse problemconvex optimizationgaussian random variables |
| spellingShingle | Valérie Gauthier-Umaña Henryk Gzyl Enrique ter Horst Decoding as a linear ill-posed problem: The entropy minimization approach AIMS Mathematics ill-posed inverse problems decoding as inverse problem convex optimization gaussian random variables |
| title | Decoding as a linear ill-posed problem: The entropy minimization approach |
| title_full | Decoding as a linear ill-posed problem: The entropy minimization approach |
| title_fullStr | Decoding as a linear ill-posed problem: The entropy minimization approach |
| title_full_unstemmed | Decoding as a linear ill-posed problem: The entropy minimization approach |
| title_short | Decoding as a linear ill-posed problem: The entropy minimization approach |
| title_sort | decoding as a linear ill posed problem the entropy minimization approach |
| topic | ill-posed inverse problems decoding as inverse problem convex optimization gaussian random variables |
| url | https://www.aimspress.com/article/doi/10.3934/math.2025192 |
| work_keys_str_mv | AT valeriegauthierumana decodingasalinearillposedproblemtheentropyminimizationapproach AT henrykgzyl decodingasalinearillposedproblemtheentropyminimizationapproach AT enriqueterhorst decodingasalinearillposedproblemtheentropyminimizationapproach |