## How to get close to the median shape (2006)

### Cached

### Download Links

Venue: | Comput. Geom. Theory Appl |

Citations: | 9 - 2 self |

### BibTeX

@ARTICLE{Har-peled06howto,

author = {Sariel Har-peled and Lewis Carol},

title = {How to get close to the median shape},

journal = {Comput. Geom. Theory Appl},

year = {2006},

volume = {36},

pages = {39--51}

}

### OpenURL

### Abstract

They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap. – The Hunting of the Snark,

### Citations

316 |
Nonlinear Regression
- Seber, Wild
- 2003
(Show Context)
Citation Context ...on algorithm for this problem [Cla05] which works via sampling. The problem seems to be harder once the shape we consider is not a linear subspace. There is considerable work on nonlinear regressions =-=[SW89]-=- (i.e., extension of the L2 least squares technique) for various shapes, but there not seems to be an efficient guaranteed approximation algorithm even for the “easy” problem of L1-fitting a circle to... |

178 | Fast Monte-Carlo algorithms for finding low-rank approximations - Frieze, Kannan, et al. |

96 | K.R.: Approximating extent measures of points
- Agarwal, Har-Peled, et al.
- 2004
(Show Context)
Citation Context ...uire O(n3 ) time. Consequently, attention has shifted to developing approximation algorithms [BH01, ZS02]. A general approximation technique was recently developed for such problems by Agarwal et al. =-=[AHV04]-=-. This implies among other things that one can approximate the circle that best fit a set of points in the plane in O(n + 1/εO(1) ) time, where the fitting measure is the maximum distance of the point... |

80 | Range searching with semi-algebraic sets
- Agarwal, Matousek
- 1994
(Show Context)
Citation Context ...linear function and gi(x, y, z) = hi(L(x, y, z)). The linearization dimension is always bounded by the number of different monomials appearing in the polynomials p1, . . . , pn. Agarwal and Matouˇsek =-=[AM94]-=- describe an algorithm that computes a linearization of the smallest dimension for a family of such polynomials. 3 Approximate L1-Fitting in One Dimension In this section, we consider the one dimensio... |

77 | Efficiently approximating the minimum-volume bounding box of a point set in three dimensions - Barequet, Har-Peled |

64 | Matrix Approximation and Projective Clustering via Volume Sampling - Deshpande, Rademacher, et al. - 2006 |

60 |
Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus
- Chan
(Show Context)
Citation Context ...n the plane in O(n + 1/εO(1) ) time, where the fitting measure is the maximum distance of the point to the circle (in fact, this special case was handled before by Agarwal et al. [AAHS00] and by Chan =-=[Cha02]-=-). The main problem with the L∞-fitting, is its sensitivity to noise and outliers. There are two natural remedies. The first is to change the target function to be less sensitive to outliers. For exam... |

47 | Coresets for k-means and k-median clustering and their applications - Har-Peled, Mazumdar - 2004 |

28 | Y.: Shape Fitting with Outliers
- Har-Peled, Wang
- 2004
(Show Context)
Citation Context ...he second approach is to specify a number k of outliers in advance and find the best shape L∞-fitting all but k of the input points. Har-Peled and Wang showed that there is a coreset for this problem =-=[HW04]-=-, and as such it can be solved ∗ Alternative titles for this paper include: “How to stay connected with your inner circle” and “How to compute one ring to rule them all”. † Department of Computer Scie... |

26 | Smaller coresets for k-median and k-means clustering. Discrete Computational Geometry, 37, 3–19. Oper Res - Har-Peled, Kushal - 2007 |

25 | Finding minimal enclosing boxes - O’Rourke - 1985 |

23 | Approximation and exact algorithms for minimum – width annuli and shells
- Agarwal, Aronov, et al.
- 1999
(Show Context)
Citation Context ...fit a set of points in the plane in O(n + 1/εO(1) ) time, where the fitting measure is the maximum distance of the point to the circle (in fact, this special case was handled before by Agarwal et al. =-=[AAHS00]-=- and by Chan [Cha02]). The main problem with the L∞-fitting, is its sensitivity to noise and outliers. There are two natural remedies. The first is to change the target function to be less sensitive t... |

11 | Algorithms for a minimum volume enclosing simplex in three dimensions - Zhou, Suri |

10 | Robust shape fitting via peeling and grating coresets
- Agarwal, Har-Peled, et al.
- 2006
(Show Context)
Citation Context ...apes. The work of Har-Peled and Wang was motivated by the aforementioned problem of L1-fitting a circle to a set of points. (The results of Har-Peled and Wang were recently improved by Agarwal et al. =-=[AHY06]-=-, but since the improvement is not significant for our purposes we will stick with the older reference.) Our Results. In this paper, we describe a general technique for computing (1 + ε)-approximate s... |

10 |
Subgradient and sampling algorithms for l1-regression
- Clarkson
- 2005
(Show Context)
Citation Context ...d using linear programming techniques, in polynomial time in high dimensions, and linear time in constant dimension [YKII88]. Recently, Clarkson gave a faster approximation algorithm for this problem =-=[Cla05]-=- which works via sampling. The problem seems to be harder once the shape we consider is not a linear subspace. There is considerable work on nonlinear regressions [SW89] (i.e., extension of the L2 lea... |

10 | Subgradient and sampling algorithms for ℓ1 regression
- Clarkson
- 2005
(Show Context)
Citation Context ...d using linear programming techniques, in polynomial time in high dimensions, and linear time in constant dimension [YKII88]. Recently, Clarkson gave a faster approximation algorithm for this problem =-=[Cla05]-=- which works via sampling. The problem seems to be harder once the shape we consider is not a linear subspace. There is considerable work on nonlinear regressions [SW89] (i.e., extension of the L2 lea... |

8 |
Algorithms for vertical and orthogonal L1 linear approximation of points
- Yamamoto, Kato, et al.
- 1988
(Show Context)
Citation Context ...ences therein. As for the L1-fitting of a linear subspace, this problem can be solved using linear programming techniques, in polynomial time in high dimensions, and linear time in constant dimension =-=[YKII88]-=-. Recently, Clarkson gave a faster approximation algorithm for this problem [Cla05] which works via sampling. The problem seems to be harder once the shape we consider is not a linear subspace. There ... |

4 |
On range-searching with semi-algebraic sets
- Agarwal, Matouˇsek
- 1993
(Show Context)
Citation Context ...r function and gi(x, y, z) = hi(L(x, y, z)). The linearization dimension 10 isis always bounded by the number of different monomials appearing in the polynomials p1, . . . , pn. Agarwal and Matouˇsek =-=[AM94]-=- describe an algorithm that computes a linearization of the smallest dimension for a family of such polynomials. 3 Approximate L1-Fitting in One Dimension In this section, we consider the one dimensio... |

4 | Analysis of incomplete data and an intrinsic-dimension helly theorem
- Gao, Langberg, et al.
(Show Context)
Citation Context ... ball b and fi). An approximation algorithm for this problem that has polynomial dependency on the dimension d (but bad dependency on the dimensions of the flats) was recently published by Gao et al. =-=[GLS06]-=-. Here, we are interested in finding the point c that minimizes the sum of distances of the point c to the flats f1, . . . , fn. Namely, this is the 1-median clustering problem for partial data. Consi... |

2 |
Approximate l1 and l2 circle fitting in (easy) polynomial time. manuscript
- Har-Peled, Koltun
- 2004
(Show Context)
Citation Context ...pproximate L∞-fitting problem. This is the first linear time algorithm for this problem. The only previous algorithm directly relevant for this result, we are aware of, is due to Har-Peled and Koltun =-=[HK04a]-=- which in O(n 2 ε −2 log 2 n) time approximates the best circle L1-fitting a set of points in the plane. The paper is organized as follows. In Section 2 we introduce some necessary preliminaries. In S... |

1 |
How to get close to the median shape. Available from http://www.uiuc.edu/~sariel/papers/05/l1_ fitting
- Har-Peled
- 2005
(Show Context)
Citation Context ...mma 3.2 Let A be a set of n real numbers, and let ψ and z be any two real numbers. We have that ˛ν (z) − |A| · |ψ − z| ˛ ≤ A ν A (ψ). Proof: Omitted. Included in the online full-version of this paper =-=[Har05]-=-. Lemma 3.3 It holds ν (z) ≈ ν (z), for any z ∈ IR. Z ε/5 S Proof: We claim that ˛ ˛ ˛˛νZ ˛ (z) − ν (z) ˛ ≤ (ε/10)ν (z), S Z for all z ∈ IR. Indeed, let τ be a median point of Z and observe that ν Z (... |

1 | Separability with outliers
- Har-Peled, Koltun
- 2005
(Show Context)
Citation Context ...pproximate L∞-fitting problem. This is the first linear time algorithm for this problem. The only previous algorithm directly relevant for this result, we are aware of, is due to Har-Peled and Koltun =-=[HK05a]-=- that, in O(n 2 ε −2 log 2 n) time, approximates the best circle L1-fitting a set of points in the plane. Comment on running time. The running time of our algorithms is O(n+poly(log n, 1/ε)) = O(n + p... |