Coresets for Vector Summarization with Applications to Network Graphs

TL;DR

This paper introduces a deterministic coreset algorithm for vector summarization that approximates the mean of large vector sets efficiently, with applications in network graph analysis and user activity summarization.

Contribution

The paper presents a novel deterministic algorithm for constructing small, weighted subsets (coresets) that approximate the mean of high-dimensional vectors, independent of data size and dimension.

Findings

The algorithm achieves an approximation error proportional to the variance of the data.

It maintains an approximate sum of vectors in streaming settings with memory independent of dimension.

Effective in identifying heavy hitters and summarizing user activity in large datasets.

Abstract

We provide a deterministic data summarization algorithm that approximates the mean $\bar{p}=\frac{1}{n}\sum_{p\in P} p$ of a set $P$ of $n$ vectors in $\REAL^d$, by a weighted mean $\tilde{p}$ of a \emph{subset} of $O(1/\eps)$ vectors, i.e., independent of both $n$ and $d$. We prove that the squared Euclidean distance between $\bar{p}$ and $\tilde{p}$ is at most $\eps$ multiplied by the variance of $P$. We use this algorithm to maintain an approximated sum of vectors from an unbounded stream, using memory that is independent of $d$, and logarithmic in the $n$ vectors seen so far. Our main application is to extract and represent in a compact way friend groups and activity summaries of users from underlying data exchanges. For example, in the case of mobile networks, we can use GPS traces to identify meetings, in the case of social networks, we can use information exchange to identify friend groups. Our algorithm provably identifies the {\it Heavy Hitter} entries in a proximity (adjacency) matrix. The Heavy Hitters can be used to extract and represent in a compact way friend groups and activity summaries of users from underlying data exchanges. We evaluate the algorithm on several large data sets.