单调和快衰减单调分布的通用信源编码
Universal Source Coding for Monotonic and Fast Decaying Monotonic Distributions
摘要：
研究了单调分布序列的普适压缩问题。我们表明，对于单调分布的大小$k$的字母表，每个概率参数的成本基本上是$0.5 \log (n/k^3)$位，其中$n$是编码序列长度，只要$k = o(n^{1/3})$。否则，对于$k = O(n)$，总的平均序列冗余度是全部$O(n^{1/3+ε})$位。然后，我们表明，存在一个子类的单调分布在无限的字母表的冗余度的$O(n^{1/3+ε})$位整体仍然是可以实现的。这个类包含快速衰减分布，包括许多整数分布和几何分布。对于一些较慢的衰减，包括其他分布的整数，冗余的$o(n)$位整体上是可实现的，其中的方法来计算特定的冗余率为这样的分布。结果是特别正确的有限熵单调分布。最后，我们研究了单个序列的冗余行为，假设一个序列是由单调分布。我们表明，序列的经验分布是单调的，个人的冗余界类似的平均情况下，可以得到。然而，即使违反了经验分布中的单调性，相对于单调最大似然描述长度减少每符号个体序列冗余仍然是可实现的。
英文摘要：
We study universal compression of sequences generated by monotonic distributions. We show that for a monotonic distribution over an alphabet of size $k$, each probability parameter costs essentially $0.5 \log (n/k^3)$ bits, where $n$ is the coded sequence length, as long as $k = o(n^{1/3})$. Otherwise, for $k = O(n)$, the total average sequence redundancy is $O(n^{1/3+ε})$ bits overall. We then show that there exists a sub-class of monotonic distributions over infinite alphabets for which redundancy of $O(n^{1/3+ε})$ bits overall is still achievable. This class contains fast decaying distributions, including many distributions over the integers and geometric distributions. For some slower decays, including other distributions over the integers, redundancy of $o(n)$ bits overall is achievable, where a method to compute specific redundancy rates for such distributions is derived. The results are specifically true for finite entropy monotonic distributions. Finally, we study individual sequence redundancy behavior assuming a sequence is governed by a monotonic distribution. We show that for sequences whose empirical distributions are monotonic, individual redundancy bounds similar to those in the average case can be obtained. However, even if the monotonicity in the empirical distribution is violated, diminishing per symbol individual sequence redundancies with respect to the monotonic maximum likelihood description length may still be achievable.
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