An Information Theoretic Condition for Perfect Reconstruction
| hal.structure.identifier | Télécom Paris | |
| hal.structure.identifier | Institut Polytechnique de Paris [IP Paris] | |
| dc.contributor.author | DELSOL, Idris | |
| hal.structure.identifier | Communications Numériques [COMNUM] | |
| hal.structure.identifier | Département Communications & Electronique [COMELEC] | |
| dc.contributor.author | RIOUL, Olivier | |
| hal.structure.identifier | Communications Numériques [COMNUM] | |
| hal.structure.identifier | Département Communications & Electronique [COMELEC] | |
| dc.contributor.author | BÉGUINOT, Julien | |
| hal.structure.identifier | Centre Sciences des Processus Industriels et Naturels [SPIN-ENSMSE] | |
| hal.structure.identifier | École des Mines de Saint-Étienne [Mines Saint-Étienne MSE] | |
| hal.structure.identifier | Laboratoire Ondes et Matière d'Aquitaine [LOMA] | |
| hal.structure.identifier | Université de Bordeaux [UB] | |
| dc.contributor.author | RABIET, Victor | |
| hal.structure.identifier | Laboratoire Instrumentation Intelligente Distribuée et Embarquée (CEA, LIST) [LIIDE (CEA, LIST)] | |
| dc.contributor.author | SOULOUMIAC, Antoine | |
| dc.date.issued | 2024-01-19 | |
| dc.identifier.issn | 1099-4300 | |
| dc.description.abstractEn | A new information theoretic condition is presented for reconstructing a discrete random variable X based on the knowledge of a set of discrete functions of X. The reconstruction condition is derived from Shannon’s 1953 lattice theory with two entropic metrics of Shannon and Rajski. Because such a theoretical material is relatively unknown and appears quite dispersed in different references, we first provide a synthetic description (with complete proofs) of its concepts, such as total, common, and complementary information. The definitions and properties of the two entropic metrics are also fully detailed and shown to be compatible with the lattice structure. A new geometric interpretation of such a lattice structure is then investigated, which leads to a necessary (and sometimes sufficient) condition for reconstructing the discrete random variable X given a set {X1,…,Xn} of elements in the lattice generated by X. Intuitively, the components X1,…,Xn of the original source of information X should not be globally “too far away” from X in the entropic distance in order that X is reconstructable. In other words, these components should not overall have too low of a dependence on X; otherwise, reconstruction is impossible. These geometric considerations constitute a starting point for a possible novel “perfect reconstruction theory”, which needs to be further investigated and improved along these lines. Finally, this condition is illustrated in five specific examples of perfect reconstruction problems: the reconstruction of a symmetric random variable from the knowledge of its sign and absolute value, the reconstruction of a word from a set of linear combinations, the reconstruction of an integer from its prime signature (fundamental theorem of arithmetic) and from its remainders modulo a set of coprime integers (Chinese remainder theorem), and the reconstruction of the sorting permutation of a list from a minimal set of pairwise comparisons. | |
| dc.language.iso | en | |
| dc.publisher | MDPI | |
| dc.subject.en | information lattice | |
| dc.subject.en | common information | |
| dc.subject.en | complementary information | |
| dc.subject.en | Rajski distance | |
| dc.subject.en | Shannon distance | |
| dc.subject.en | dependency coefficient | |
| dc.subject.en | relative redundancy | |
| dc.subject.en | convex envelope | |
| dc.subject.en | perfect reconstruction | |
| dc.subject.en | Signal processing | |
| dc.title.en | An Information Theoretic Condition for Perfect Reconstruction | |
| dc.type | Article de revue | |
| dc.identifier.doi | 10.3390/e26010086 | |
| dc.subject.hal | Informatique [cs]/Traitement du signal et de l'image | |
| dc.subject.hal | Mathématiques [math]/Géométrie métrique [math.MG] | |
| bordeaux.journal | Entropy | |
| bordeaux.page | 86 | |
| bordeaux.volume | 26 | |
| bordeaux.issue | 1 | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-04416099 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-04416099v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Entropy&rft.date=2024-01-19&rft.volume=26&rft.issue=1&rft.spage=86&rft.epage=86&rft.eissn=1099-4300&rft.issn=1099-4300&rft.au=DELSOL,%20Idris&RIOUL,%20Olivier&B%C3%89GUINOT,%20Julien&RABIET,%20Victor&SOULOUMIAC,%20Antoine&rft.genre=article |
Fichier(s) constituant ce document
| Fichiers | Taille | Format | Vue |
|---|---|---|---|
|
Il n'y a pas de fichiers associés à ce document. |
|||