In filtering, each output is produced by a certain number of different inputs. We explore the statistics of this degeneracy in an explicitly treatable filtering problem in which filtering performs the maximal compression of relevant information contained in inputs (arrays of zeros and ones). This problem serves as a reference model for the statistics of filtering and related sampling problems. The filter patterns in this problem conveniently allow a microscopic, combinatorial consideration. This allows us to find the statistics of outputs, namely the exact distribution of output degeneracies, for arbitrary input sizes. We observe that the resulting degeneracy distribution of outputs decays as $e^{-c{\log }^{\alpha }d}$ with degeneracy $d$, where $c$ is a constant and exponent $\alpha >1$, i.e., faster than a power law. Importantly, its form essentially depends on the size of the input dataset, appearing to be closer to a power-law dependence for small dataset sizes than for large ones. We demonstrate that for sufficiently small input dataset sizes typical for empirical studies, this distribution could be easily perceived as a power law. We extend our results to filter patterns of various sizes and demonstrate that the shortest filter pattern provides the maximum informative representations of the inputs.

中文翻译：

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过滤和压缩中出现的复杂分布

在滤波中，每个输出由一定数量的不同输入产生。我们在可显式处理的过滤问题中探索这种退化的统计信息，其中过滤对输入（零和一的数组）中包含的相关信息进行最大压缩。该问题可作为过滤和相关抽样问题统计信息的参考模型。这个问题中的过滤器图案方便地允许微观的，组合的考虑。这使我们能够找到任意输入大小的输出统计信息，即输出简并性的确切分布。我们观察到，输出的简并分布会随着${Ë}^{-C{\mathrm{日志}}^{\alpha }d}$ 简并 $d$，在哪里 $C$ 是一个常数和指数 $\alpha >\mathrm{1个}$即比幂律更快。重要的是，其形式基本上取决于输入数据集的大小，对于大数据集而言，看起来似乎更接近于幂律依赖性。我们证明，对于经验研究典型的足够小的输入数据集大小，可以很容易地将此​​分布视为幂定律。我们将结果扩展到各种大小的过滤器模式，并证明最短的过滤器模式可提供最大程度的信息表示。