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DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.62

Streaming Coreset Constructions for M-Estimators

Authors: Vladimir Braverman, Dan Feldman, Harry Lang, and Daniela Rus

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract

We introduce a new method of maintaining a (k,epsilon)-coreset for clustering M-estimators over insertion-only streams. Let (P,w) be a weighted set (where w : P - > [0,infty) is the weight function) of points in a rho-metric space (meaning a set X equipped with a positive-semidefinite symmetric function D such that D(x,z) <=rho(D(x,y) + D(y,z)) for all x,y,z in X). For any set of points C, we define COST(P,w,C) = sum_{p in P} w(p) min_{c in C} D(p,c). A (k,epsilon)-coreset for (P,w) is a weighted set (Q,v) such that for every set C of k points, (1-epsilon)COST(P,w,C) <= COST(Q,v,C) <= (1+epsilon)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 psi(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 (psi(x) =x) and k-means (psi(x) = x^2) 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(epsilon^{-2} k log k log n) points of storage. The previous state-of-the-art required storing at least O(epsilon^{-2} k log k log^{4} n) points.

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Vladimir Braverman, Dan Feldman, Harry Lang, and Daniela Rus. Streaming Coreset Constructions for M-Estimators. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{braverman_et_al:LIPIcs.APPROX-RANDOM.2019.62,
  author =	{Braverman, Vladimir and Feldman, Dan and Lang, Harry and Rus, Daniela},
  title =	{{Streaming Coreset Constructions for M-Estimators}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{62:1--62:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.62},
  URN =		{urn:nbn:de:0030-drops-112778},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.62},
  annote =	{Keywords: Streaming, Clustering, Coresets}
}

Document

Invited Talk

DOI: 10.4230/LIPIcs.SoCG.2016.1

Toward Pervasive Robots (Invited Talk)

Authors: Daniela Rus

Published in: LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Abstract

The digitization of practically everything coupled with the mobile Internet, the automation of knowledge work, and advanced robotics promises a future with democratized use of machines and wide-spread use of robots and customization. However, pervasive use of robots remains a hard problem. Where are the gaps that we need to address in order to advance toward a future where robots are common in the world and they help reliably with physical tasks? What is the role of geometric reasoning along this trajectory? In this talk I will discuss challenges toward pervasive use of robots and recent developments in geometric algorithms for customizing robots. I will focus on a suite of gemetric algorithms for automatically designing, fabricating, and tasking robots using a print-and-fold approach. I will also describe how geometric reasoning can play a role in creating robots more capable of reasoning in the world. By enabling on-demand creation of programmable robots, we can begin to imagine a world with one robot for every physical task.

Cite as

Daniela Rus. Toward Pervasive Robots (Invited Talk). In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{rus:LIPIcs.SoCG.2016.1,
  author =	{Rus, Daniela},
  title =	{{Toward Pervasive Robots}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{1:1--1:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{Fekete, S\'{a}ndor and Lubiw, Anna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.1},
  URN =		{urn:nbn:de:0030-drops-58939},
  doi =		{10.4230/LIPIcs.SoCG.2016.1},
  annote =	{Keywords: rus@csail.mit.edu}
}

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Documents authored by Rus, Daniela

Streaming Coreset Constructions for M-Estimators

Abstract

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Toward Pervasive Robots (Invited Talk)

Abstract

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