置换空间中的压缩
Compression in the Space of Permutations
摘要：
我们研究了以置换形式的数据的有损压缩（信源编码）。这个问题在有序数据或排序的存储以及排序算法的分析中有直接的应用。分析了均匀分布下置换空间的率失真特性，以及恢复后允许有界失真的最小可达压缩率。我们的分析是关于不同的实际和有用的失真措施，包括肯德尔-陶距离，斯皮尔曼的foottrule，切比雪夫距离和反演- $\ell_1$距离。我们建立等价的源代码设计在一定的失真，并显示简单的显式代码设计，产生低编码/解码的复杂性和渐近最优。最后，我们表明，对于Mallow模型，一个流行的非均匀排列模型的置换空间，熵和最大失真在零速率是远远低于均匀的同行，这激励了未来的设计高效的压缩方案，这种模式。
英文摘要：
We investigate lossy compression (source coding) of data in the form of permutations. This problem has direct applications in the storage of ordinal data or rankings, and in the analysis of sorting algorithms. We analyze the rate-distortion characteristic for the permutation space under the uniform distribution, and the minimum achievable rate of compression that allows a bounded distortion after recovery. Our analysis is with respect to different practical and useful distortion measures, including Kendall-tau distance, Spearman's footrule, Chebyshev distance and inversion-$\ell_1$ distance. We establish equivalence of source code designs under certain distortions and show simple explicit code designs that incur low encoding/decoding complexities and are asymptotically optimal. Finally, we show that for the Mallows model, a popular nonuniform ranking model on the permutation space, both the entropy and the maximum distortion at zero rate are much lower than the uniform counterparts, which motivates the future design of efficient compression schemes for this model.
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