Compressed Counting

TL;DR

Compressed Counting introduces an efficient method for estimating the pth frequency moment in data streams, especially near p=1, using skewed stable projections to reduce sample complexity.

Contribution

It proposes a novel technique called skewed stable random projections for efficiently computing frequency moments in data streams with minimal sample complexity.

Findings

Sample complexity is O(1/ε) for p close to 1.

Applicable to Turnstile data streams with non-negative entries.

Provides algorithms for estimating logarithmic norms and distances.

Abstract

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+/\Delta with small \Delta, the sample complexity of a counter system should be low (continuously as a function of \Delta). We show at small \Delta the sample complexity (number of projections) k = O(1/\epsilon) instead of O(1/\epsilon^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.