Optimal vector quantization in terms of Wasserstein distance
- The optimal quantizer in memory-size constrained vector quantization induces a quantization error which is equal to a Wasserstein distortion. However, for the optimal (Shannon-)entropy constrained quantization error a proof for a similar identity is still missing. Relying on principal results of the optimal mass transportation theory, we will prove that the optimal quantization error is equal to a Wasserstein distance. Since we will state the quantization problem in a very general setting, our approach includes the R\'enyi-$\alpha$-entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained $(\alpha = 1)$ and memory-size constrained $(\alpha = 0)$ quantization. Additionally, we will derive for certain distance functions codecell convexity for quantizers with a finite codebook. Using other methods, this regularity in codecell geometry has already been proved earlier by Gy\"{o}rgy and Linder.
| Author: | Wolfgang Kreitmeier |
|---|---|
| URN: | urn:nbn:de:bvb:739-opus-22502 |
| Document Type: | Preprint |
| Language: | English |
| Year of Completion: | 2011 |
| Date of Publication (online): | 2011/04/18 |
| Publishing Institution: | Universität Passau |
| Release Date: | 2011/04/18 |
| Tag: | R\'enyi-$\alpha$-entropy; Wasserstein distance; codecell convexity; optimal quantization error |
| GND Keyword: | Maßtheorie; Transporttheorie; Quantisierung; Entropie |
| Note: | This is a preprint of an article accepted for publication in the Journal of Multivariate Analysis ISSN 0047-259X. The original publication is available at http://www.elsevier.com/. The digital object identifier (DOI) of the definitive article is 10.1016/j.jmva.2011.04.005. |
| Source: | Journal of Multivariate Analysis. ISSN 0047-259X |
| Institutes: | Fakultät für Informatik und Mathematik / Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik |
| Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
| open_access (DINI-Set): | open_access |
| Licence (German): | Standardbedingung laut Einverständniserklärung |

