极值堆是速率无关函数的最小充分统计量，具有科尔莫戈罗夫最优压缩性。
The Extremum Stack is a Minimal Sufficient Statistic for Rate-Independent Functionals: A Kolmogorov Complexity Characterisation
- 用极值堆表示离散序列，可完整保留所有可计算因果速率无关函数的信息。
- 压缩后所需最少比特数为K(Π_n) - O(1)，且该下界可被堆结构达到。
- 适用于需保持完整函数类响应的系统建模与高效数据压缩场景。
我们证明了离散序列的极值堆是所有可计算、因果、速率无关函数类的最小充分统计量，基于科尔莫戈罗夫复杂度。具体地，建立了K(Pi_n) - O(1) ≤ K_R(u_{0:n}) ≤ K(Pi_n) + O(1)，其中K_R(u_{0:n})为回答该函数类中所有查询的最短程序长度，O(1)项不依赖序列长度n或堆深度k。充分性源于普莱斯亚赫滞回算子的经典抹除性质；最小性通过显式验证其速率不变性的有限指示族建立。任何保留完整函数类R的滞回流压缩，至少需保留K(Pi_n) - O(1)比特；由此结果导出的堆式压缩算法具备科尔莫戈罗夫最优性，这是标准时间序列压缩方法无法提供的。
原文摘要 · Abstract (English)
We prove that the extremum stack of a discrete sequence is a minimal sufficient statistic for the class of all computable, causal, rate-independent functionals, in the sense of Kolmogorov complexity. Specifically, we establish K(Pi_n) - O(1) <= K_R(u_{0:n}) <= K(Pi_n) + O(1), where K_R(u_{0:n}) is the length of the shortest program answering every query in the class R, and the O(1) overhead is independent of both the sequence length n and the stack depth k. Sufficiency follows from the classical wiping property of the Preisach hysteresis operator. Minimality is established via a finite indicator family whose rate-independence is verified explicitly. Any compression of a hysteresis-driven stream that preserves the full class R must therefore retain at least K(Pi_n) - O(1) bits; the stack-based compression algorithm implied by the result carries a Kolmogorov optimality guarantee that none of the standard time-series compression methods provide.
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