Streaming coreset constructions for M-estimators
Name
LIPIcs-APPROX-RANDOM-2019-62.pdf
Description
Published version
Size
538.32 KB
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Unknown
Checksum (MD5)
7c7f414cbd3026b6a7b6fcbb30bd461b
Author(s)
Braverman, Vladimir
Feldman, Dan
Lang, Harry
Rus, Daniela L
Date Issued
September 2019
Journal
Leibniz International Proceedings in Informatics, LIPIcs
Citation
2019. "Streaming coreset constructions for M-estimators." Leibniz International Proceedings in Informatics, LIPIcs, 145.
Version
Final published version
Abstract
© Vladimir Braverman, Dan Feldman, Harry Lang, and Daniela Rus. We introduce a new method of maintaining a (k, ϵ)-coreset for clustering M-estimators over insertion-only streams. Let (P, w) be a weighted set (where w : P → [0, ∞) is the weight function) of points in a ρ-metric space (meaning a set X equipped with a positive-semidefinite symmetric function D such that D(x, z) ≤ ρ(D(x, y) + D(y, z)) for all x, y, z ∈ X). For any set of points C, we define COST(P, w, C) = ∑p∈P w(p) minc∈C D(p, c). A (k, ϵ)-coreset for (P, w) is a weighted set (Q, v) such that for every set C of k points, (1 − ϵ)COST(P, w, C) ≤ COST(Q, v, C) ≤ (1 + ϵ)COST(P, w, C). Essentially, the coreset (Q, v) can be used in place of (P, w) for all operations concerning the COST function. Coresets, as a method of data reduction, are used to solve fundamental problems in machine learning of streaming and distributed data. M-estimators are functions D(x, y) that can be written as ψ(d(x, y)) where (X, d) is a true metric (i.e. 1-metric) space. Special cases of M-estimators include the well-known k-median (ψ(x) = x) and k-means (ψ(x) = x2) functions. Our technique takes an existing offline construction for an M-estimator coreset and converts it into the streaming setting, where n data points arrive sequentially. To our knowledge, this is the first streaming construction for any M-estimator that does not rely on the merge-and-reduce tree. For example, our coreset for streaming metric k-means uses O(ϵ−2k log k log n) points of storage. The previous state-of-the-art required storing at least O(ϵ−2k log k log4 n) points.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Terms of Use
Creative Commons Attribution 4.0 International license
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.62
