Results on the Redundancy of Universal Compression for Finite-Length Sequences

Ahmad Beirami; Faramarz Fekri

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arXiv · 1110.5710

Results on the Redundancy of Universal Compression for Finite-Length Sequences

Ahmad Beirami·Faramarz Fekri

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Abstract

In this paper, we investigate the redundancy of universal coding schemes on smooth parametric sources in the finite-length regime. We derive an upper bound on the probability of the event that a sequence of length $n$, chosen using Jeffreys' prior from the family of parametric sources with $d$ unknown parameters, is compressed with a redundancy smaller than $(1-ε)\frac{d}{2}\log n$ for any $ε>0$. Our results also confirm that for large enough $n$ and $d$, the average minimax redundancy provides a good estimate for the redundancy of most sources. Our result may be used to evaluate the performance of universal source coding schemes on finite-length sequences. Additionally, we precisely characterize the minimax redundancy for two--stage codes. We demonstrate that the two--stage assumption incurs a negligible redundancy especially when the number of source parameters is large. Finally, we show that the redundancy is significant in the compression of small sequences.

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Ahmad Beirami, Faramarz Fekri. 2011-10-26. Results on the Redundancy of Universal Compression for Finite-Length Sequences. https://arxiv.org/abs/1110.5710

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