Consistent Subset Sampling

TL;DR

This paper introduces an efficient method for consistent subset sampling in large datasets, optimizing time and space complexity, and demonstrates its application in data mining tasks like frequent itemset and bipartite clique estimation.

Contribution

It generalizes consistent sampling to size-k subsets with improved time and space complexity using a novel hash-based approach.

Findings

Achieves expected time complexity of Θ(b^{⌈k/2⌉} log log b + pb^k)

Uses space complexity of Θ(b^{⌈k/4⌉})

Effectively estimates frequent itemsets and bipartite cliques in data streams

Abstract

Consistent sampling is a technique for specifying, in small space, a subset $S$ of a potentially large universe $U$ such that the elements in $S$ satisfy a suitably chosen sampling condition. Given a subset $\mathcal{I}\subseteq U$ it should be possible to quickly compute $\mathcal{I}\cap S$, i.e., the elements in $\mathcal{I}$ satisfying the sampling condition. Consistent sampling has important applications in similarity estimation, and estimation of the number of distinct items in a data stream. In this paper we generalize consistent sampling to the setting where we are interested in sampling size-$k$ subsets occurring in some set in a collection of sets of bounded size $b$, where $k$ is a small integer. This can be done by applying standard consistent sampling to the $k$-subsets of each set, but that approach requires time $\Theta(b^k)$. Using a carefully designed hash function, for a given sampling probability $p \in (0,1]$, we show how to improve the time complexity to $\Theta(b^{\lceil k/2\rceil}\log \log b + pb^k)$ in expectation, while maintaining strong concentration bounds for the sample. The space usage of our method is $\Theta(b^{\lceil k/4\rceil})$. We demonstrate the utility of our technique by applying it to several well-studied data mining problems. We show how to efficiently estimate the number of frequent $k$-itemsets in a stream of transactions and the number of bipartite cliques in a graph given as incidence stream. Further, building upon a recent work by Campagna et al., we show that our approach can be applied to frequent itemset mining in a parallel or distributed setting. We also present applications in graph stream mining.