Coresets for Vector Summarization with Applications to Network Graphs

Dan Feldman, Sedat Ozer, Daniela Rus
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:1117-1125, 2017.

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 $\mathbb{R}^d$, by a weighted mean $\tilde{p}$ of a subset of $O(1/\epsilon)$ 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 $\epsilon$ 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 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v70-feldman17a, title = {Coresets for Vector Summarization with Applications to Network Graphs}, author = {Dan Feldman and Sedat Ozer and Daniela Rus}, booktitle = {Proceedings of the 34th International Conference on Machine Learning}, pages = {1117--1125}, year = {2017}, editor = {Precup, Doina and Teh, Yee Whye}, volume = {70}, series = {Proceedings of Machine Learning Research}, month = {06--11 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v70/feldman17a/feldman17a.pdf}, url = {https://proceedings.mlr.press/v70/feldman17a.html}, 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 $\mathbb{R}^d$, by a weighted mean $\tilde{p}$ of a subset of $O(1/\epsilon)$ 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 $\epsilon$ 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 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.} }
Endnote
%0 Conference Paper %T Coresets for Vector Summarization with Applications to Network Graphs %A Dan Feldman %A Sedat Ozer %A Daniela Rus %B Proceedings of the 34th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2017 %E Doina Precup %E Yee Whye Teh %F pmlr-v70-feldman17a %I PMLR %P 1117--1125 %U https://proceedings.mlr.press/v70/feldman17a.html %V 70 %X 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 $\mathbb{R}^d$, by a weighted mean $\tilde{p}$ of a subset of $O(1/\epsilon)$ 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 $\epsilon$ 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 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.
APA
Feldman, D., Ozer, S. & Rus, D.. (2017). Coresets for Vector Summarization with Applications to Network Graphs. Proceedings of the 34th International Conference on Machine Learning, in Proceedings of Machine Learning Research 70:1117-1125 Available from https://proceedings.mlr.press/v70/feldman17a.html.

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