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How compressible are innovation processes?

Ghourchian, H ; Sharif University of Technology | 2018

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  1. Type of Document: Article
  2. DOI: 10.1109/TIT.2018.2822660
  3. Publisher: Institute of Electrical and Electronics Engineers Inc , 2018
  4. Abstract:
  5. The sparsity and compressibility of finite-dimensional signals are of great interest in fields, such as compressed sensing. The notion of compressibility is also extended to infinite sequences of independent identically distributed or ergodic random variables based on the observed error in their nonlinear $k$ -term approximation. In this paper, we use the entropy measure to study the compressibility of continuous-domain innovation processes (alternatively known as white noise). Specifically, we define such a measure as the entropy limit of the doubly quantized (time and amplitude) process. This provides a tool to compare the compressibility of various innovation processes. It also allows us to identify an analogue of the concept of 'entropy dimension' which was originally defined by Rényi for random variables. Particular attention is given to stable and impulsive Poisson innovation processes. Here, our results recognize Poisson innovations as the more compressible ones with an entropy measure far below that of stable innovations. While this result departs from the previous knowledge regarding the compressibility of impulsive Poisson laws compared with continuous fat-tailed distributions, our entropy measure ranks $alpha$ -stable innovations according to their tail. © 1963-2012 IEEE
  6. Keywords:
  7. Impulsive Poisson process ; Stable innovation ; White Lévy noise ; Compressibility ; Entropy ; Poisson distribution ; Random variables ; Continuous domain ; Entropy dimension ; Entropy limit ; Entropy measure ; Fat-tailed distributions ; Finite dimensional ; Innovation process ; Poisson process ; White noise
  8. Source: IEEE Transactions on Information Theory ; Volume 64, Issue 7 , 2018 , Pages 4843-4871 ; 00189448 (ISSN)
  9. URL: https://ieeexplore.ieee.org/document/8331128