压缩计数

摘要：

计数是计算中最基本的操作之一。例如，在理论计算机科学、数据库和数据采掘中，计算p阶频率矩一直是非常活跃的研究领域。当p=1时，任务（即，对和进行计数）可以使用简单的计数器来完成。压缩计数（CC）被提出用于有效地计算数据流信号A_t的第p个频率矩，其中0<p<=2。如果流数据遵循旋转栅门模型，则CC是适用的，其中限制是在用于评估的时间t处，A_t[i]>= 0，其包括严格旋转栅门模型作为特殊情况。对于在实践中遇到的自然数据流，这种限制是次要的。CC的基本技术是我们称之为倾斜稳定随机投影的技术，它捕捉了这样的直觉：当p=1时，简单的计数器就足够了，当p = 1+/Δ且Δ较小时，计数器系统的样本复杂度应该很低（连续地作为Δ的函数）。我们证明了在小Δ时样本复杂度（投影数）k = O（1/ε）而不是O（1/ε^2）。压缩计数可以为统计和计算中的其他任务提供基本构建块，例如，数据流的估计熵，使用矩和最大似然方法的参数估计。最后，另一个贡献是一个算法近似的对数范数，\sum_{i=1}^D \log A_t[i]，和对数距离。对数距离在重尾数据的机器学习实践中很有用。

英文摘要：

Counting is among the most fundamental operations in computing. For example, counting the pth frequency moment has been a very active area of research, in theoretical computer science, databases, and data mining. When p=1, the task (i.e., counting the sum) can be accomplished using a simple counter. Compressed Counting (CC) is proposed for efficiently computing the pth frequency moment of a data stream signal A_t, where 0<p<=2. CC is applicable if the streaming data follow the Turnstile model, with the restriction that at the time t for the evaluation, A_t[i]>= 0, which includes the strict Turnstile model as a special case. For natural data streams encountered in practice, this restriction is minor. The underly technique for CC is what we call skewed stable random projections, which captures the intuition that, when p=1 a simple counter suffices, and when p = 1+/Δwith small Δ, the sample complexity of a counter system should be low (continuously as a function of Δ). We show at small Δthe sample complexity (number of projections) k = O(1/ε) instead of O(1/ε^2). Compressed Counting can serve a basic building block for other tasks in statistics and computing, for example, estimation entropies of data streams, parameter estimations using the method of moments and maximum likelihood. Finally, another contribution is an algorithm for approximating the logarithmic norm, \sum_{i=1}^D\log A_t[i], and logarithmic distance. The logarithmic distance is useful in machine learning practice with heavy-tailed data.

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