基于Kolmogorov复杂度的个体数据率失真与去噪
Rate Distortion and Denoising of Individual Data Using Kolmogorov complexity
摘要：
我们从Kolmogorov复杂性的角度研究了畸变球家族的结构。特别注意的是，返回的最小Kolmogorov复杂性的所有失真球包含一个词的约束，其基数的源字的典型率失真函数。该规范率失真函数与针对给定失真度量的更标准算法率失真函数相关。例如列表失真，汉明失真和欧几里德失真。算法率失真函数可以表现得与香农的率失真函数不同。为此，我们表明，典型的率失真函数可以，并承担了广泛的一类形状（不像香农的）;我们将低算法互信息与低Kolmogorov复杂度联系起来（并因此表明香农率失真函数的互信息公式的某些方面与使用算法互信息的类似公式表现不同）;我们探讨了包含给定单词的低Kolmogorov复杂度失真球捕获该单词的有趣特性的概念（这在香农理论中很难形式化），这提出了一种去噪方法;最后，我们表明，在对源字进行平均之后，各个源字的率失真曲线的不同行为在一定程度上消失。
英文摘要：
We examine the structure of families of distortion balls from the perspective of Kolmogorov complexity. Special attention is paid to the canonical rate-distortion function of a source word which returns the minimal Kolmogorov complexity of all distortion balls containing that word subject to a bound on their cardinality. This canonical rate-distortion function is related to the more standard algorithmic rate-distortion function for the given distortion measure. Examples are given of list distortion, Hamming distortion, and Euclidean distortion. The algorithmic rate-distortion function can behave differently from Shannon's rate-distortion function. To this end, we show that the canonical rate-distortion function can and does assume a wide class of shapes (unlike Shannon's); we relate low algorithmic mutual information to low Kolmogorov complexity (and consequently suggest that certain aspects of the mutual information formulation of Shannon's rate-distortion function behave differently than would an analogous formulation using algorithmic mutual information); we explore the notion that low Kolmogorov complexity distortion balls containing a given word capture the interesting properties of that word (which is hard to formalize in Shannon's theory) and this suggests an approach to denoising; and, finally, we show that the different behavior of the rate-distortion curves of individual source words to some extent disappears after averaging over the source words.
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