### Abstract

In this paper, two methods for one-dimensional reduction of data by hyperplane fitting are proposed. One is least α-percentile of squares, which is an extension of least median of squares estimation and minimizes the α-percentile of squared Euclidean distance. The other is least k-th power deviation, which is an extension of least squares estimation and minimizes the k-th power deviation of squared Euclidean distance. Especially, for least k-th power deviation of 0 < k ≤ 1, it is proved that a useful property, called optimal sampling property, holds in one-dimensional reduction of data by hyperplane fitting. The optimal sampling property is that the global optimum for affine hyperplane fitting passes through N data points when an -dimensional hyperplane is fitted to the N-dimensional data. The performance of the proposed methods is evaluated by line fitting to artificial data and a real image.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Pages | 278-285 |

Number of pages | 8 |

Volume | 6854 LNCS |

Edition | PART 1 |

DOIs | |

Publication status | Published - 2011 |

Event | 14th International Conference on Computer Analysis of Images and Patterns, CAIP 2011 - Seville Duration: 2011 Aug 29 → 2011 Aug 31 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 1 |

Volume | 6854 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 14th International Conference on Computer Analysis of Images and Patterns, CAIP 2011 |
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City | Seville |

Period | 11/8/29 → 11/8/31 |

### Fingerprint

### Keywords

- hyperplane fitting
- least α-percentile of squares
- least k-th power deviations
- optimal sampling property
- random sampling

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(PART 1 ed., Vol. 6854 LNCS, pp. 278-285). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6854 LNCS, No. PART 1). https://doi.org/10.1007/978-3-642-23672-3_34

**Robust hyperplane fitting based on k-th power deviation and α-quantile.** / Fujiki, Jun; Akaho, Shotaro; Hino, Hideitsu; Murata, Noboru.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*PART 1 edn, vol. 6854 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), no. PART 1, vol. 6854 LNCS, pp. 278-285, 14th International Conference on Computer Analysis of Images and Patterns, CAIP 2011, Seville, 11/8/29. https://doi.org/10.1007/978-3-642-23672-3_34

}

TY - GEN

T1 - Robust hyperplane fitting based on k-th power deviation and α-quantile

AU - Fujiki, Jun

AU - Akaho, Shotaro

AU - Hino, Hideitsu

AU - Murata, Noboru

PY - 2011

Y1 - 2011

N2 - In this paper, two methods for one-dimensional reduction of data by hyperplane fitting are proposed. One is least α-percentile of squares, which is an extension of least median of squares estimation and minimizes the α-percentile of squared Euclidean distance. The other is least k-th power deviation, which is an extension of least squares estimation and minimizes the k-th power deviation of squared Euclidean distance. Especially, for least k-th power deviation of 0 < k ≤ 1, it is proved that a useful property, called optimal sampling property, holds in one-dimensional reduction of data by hyperplane fitting. The optimal sampling property is that the global optimum for affine hyperplane fitting passes through N data points when an -dimensional hyperplane is fitted to the N-dimensional data. The performance of the proposed methods is evaluated by line fitting to artificial data and a real image.

AB - In this paper, two methods for one-dimensional reduction of data by hyperplane fitting are proposed. One is least α-percentile of squares, which is an extension of least median of squares estimation and minimizes the α-percentile of squared Euclidean distance. The other is least k-th power deviation, which is an extension of least squares estimation and minimizes the k-th power deviation of squared Euclidean distance. Especially, for least k-th power deviation of 0 < k ≤ 1, it is proved that a useful property, called optimal sampling property, holds in one-dimensional reduction of data by hyperplane fitting. The optimal sampling property is that the global optimum for affine hyperplane fitting passes through N data points when an -dimensional hyperplane is fitted to the N-dimensional data. The performance of the proposed methods is evaluated by line fitting to artificial data and a real image.

KW - hyperplane fitting

KW - least α-percentile of squares

KW - least k-th power deviations

KW - optimal sampling property

KW - random sampling

UR - http://www.scopus.com/inward/record.url?scp=80052792975&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052792975&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-23672-3_34

DO - 10.1007/978-3-642-23672-3_34

M3 - Conference contribution

AN - SCOPUS:80052792975

SN - 9783642236716

VL - 6854 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 278

EP - 285

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -