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A coreset for a set of points is a small subset of weighted points thatapproximately preserves important properties of the original set. Specifically,if $P$ is a set of points, $Q$ is a set of queries, and $f:P\timesQ\to\mathbb{R}$ is a cost function, then a set $S\subseteq P$ with weights$w:P\to[0,\infty)$ is an $\epsilon$-coreset for some parameter $\epsilon>0$ if$\sum_{s\in S}w(s)f(s,q)$ is a $(1+\epsilon)$ multiplicative approximation to$\sum_{p\in P}f(p,q)$ for all $q\in Q$. Coresets are used to solve fundamentalproblems in machine learning under various big data models of computation. Manyof the suggested coresets in the recent decade used, or could have used ageneral framework for constructing coresets whose size depends quadratically onwhat is known as total sensitivity $t$.  In this paper we improve this bound from $O(t^2)$ to $O(t\log t)$. Thus ourresults imply more space efficient solutions to a number of problems, includingprojective clustering, $k$-line clustering, and subspace approximation.Moreover, we generalize the notion of sensitivity sampling for sup-samplingthat supports non-multiplicative approximations, negative cost functions andmore. The main technical result is a generic reduction to the sample complexityof learning a class of functions with bounded VC dimension. We show thatobtaining an $(\nu,\alpha)$-sample for this class of functions with appropriateparameters $\nu$ and $\alpha$ suffices to achieve space efficient$\epsilon$-coresets.  Our result implies more efficient coreset constructions for a number ofinteresting problems in machine learning; we show applications to$k$-median/$k$-means, $k$-line clustering, $j$-subspace approximation, and theinteger $(j,k)$-projective clustering problem.

New Frameworks for Offline and Streaming Coreset Constructions