Frame expansions with erasures: an approach through the non-commutative operator theory
| Autor: | Roman Vershynin |
|---|---|
| Rok vydání: | 2005 |
| Předmět: |
math.NA
math.FA 02 engineering and technology Source coding 01 natural sciences 46B09 47B10 94A12 42C15 Position (vector) FOS: Mathematics Physical Sciences and Mathematics 0202 electrical engineering electronic engineering information engineering Oversampling Mathematics - Numerical Analysis 0101 mathematics Time complexity Commutative property Mathematics Discrete mathematics Basis (linear algebra) Applied Mathematics Linear reconstruction 010102 general mathematics Frame (networking) 020206 networking & telecommunications Numerical Analysis (math.NA) Operator theory Orthogonal basis Functional Analysis (math.FA) Mathematics - Functional Analysis Frames Overcomplete representations Multiple descriptions |
| Zdroj: | Vershynin, Roman. (2004). Frame expansions with erasures: an approach through the non-commutative operator theory. Applied and Computational Harmonic Analysis 18 (2005), 167--176. UC Davis: Department of Mathematics. Retrieved from: http://www.escholarship.org/uc/item/7qx410wv |
| ISSN: | 1063-5203 |
| DOI: | 10.1016/j.acha.2004.12.001 |
| Popis: | In modern communication systems such as the Internet, random losses of information can be mitigated by oversampling the source. This is equivalent to expanding the source using overcomplete systems of vectors (frames), as opposed to the traditional basis expansions. Dependencies among the coefficients in frame expansions often allow for better performance comparing to bases under random losses of coefficients. We show that for any n-dimensional frame, any source can be linearly reconstructed from only (n log n) randomly chosen frame coefficients, with a small error and with high probability. Thus every frame expansion withstands random losses better (for worst case sources) than the orthogonal basis expansion, for which the (n log n) bound is attained. The proof reduces to M.Rudelson's selection theorem on random vectors in the isotropic position, which is based on the non-commutative Khinchine's inequality. 12 pages |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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