### Abstract

Language | English |
---|---|

Pages | 1009-1016 |

Number of pages | 8 |

Journal | Image and Vision Computing |

Volume | 20 |

Issue number | 13-14 |

DOIs | |

Status | Published - 1 Dec 2002 |

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### Keywords

- eigenvalue decomposition
- singular value decomposition
- dynamic updating and downdating
- gaussian mixture models

### Cite this

**Adding and subtracting eigenspaces with eigenvalue decomposition and singular value decomposition.** / Hall, Peter; Marshall, David; Martin, Ralph.

Research output: Contribution to journal › Article

*Image and Vision Computing*, vol. 20, no. 13-14, pp. 1009-1016. DOI: 10.1016/S0262-8856(02)00114-2

}

TY - JOUR

T1 - Adding and subtracting eigenspaces with eigenvalue decomposition and singular value decomposition

AU - Hall,Peter

AU - Marshall,David

AU - Martin,Ralph

N1 - ID number: ISI:000180216300008

PY - 2002/12/1

Y1 - 2002/12/1

N2 - This paper provides algorithms for adding and subtracting eigenspaces, thus allowing for incremental updating and downdating of data models. Importantly, and unlike previous work, we keep an accurate track of the mean of the data, which allows our methods to be used in classification applications. The result of adding eigenspaces, each made from a set of data, is an approximation to that which would obtain were the sets of data taken together. Subtracting eigenspaces yields a result approximating that which would obtain were a subset of data used. Using our algorithms, it is possible to perform 'arithmetic' on eigenspaces without reference to the original data. Eigenspaces can be constructed using either eigenvalue decomposition (EVD) or singular value decomposition (SVD). We provide addition operators for both methods, but subtraction for EVD only, arguing there is no closed-form solution for SVD. The methods and discussion surrounding SVD provide the principle novelty in this paper. We illustrate the use of our algorithms in three generic applications, including the dynamic construction of Gaussian mixture models.

AB - This paper provides algorithms for adding and subtracting eigenspaces, thus allowing for incremental updating and downdating of data models. Importantly, and unlike previous work, we keep an accurate track of the mean of the data, which allows our methods to be used in classification applications. The result of adding eigenspaces, each made from a set of data, is an approximation to that which would obtain were the sets of data taken together. Subtracting eigenspaces yields a result approximating that which would obtain were a subset of data used. Using our algorithms, it is possible to perform 'arithmetic' on eigenspaces without reference to the original data. Eigenspaces can be constructed using either eigenvalue decomposition (EVD) or singular value decomposition (SVD). We provide addition operators for both methods, but subtraction for EVD only, arguing there is no closed-form solution for SVD. The methods and discussion surrounding SVD provide the principle novelty in this paper. We illustrate the use of our algorithms in three generic applications, including the dynamic construction of Gaussian mixture models.

KW - eigenvalue decomposition

KW - singular value decomposition

KW - dynamic updating and downdating

KW - gaussian mixture models

UR - http://dx.doi.org/10.1016/S0262-8856(02)00114-2

U2 - 10.1016/S0262-8856(02)00114-2

DO - 10.1016/S0262-8856(02)00114-2

M3 - Article

VL - 20

SP - 1009

EP - 1016

JO - Image and Vision Computing

T2 - Image and Vision Computing

JF - Image and Vision Computing

SN - 0262-8856

IS - 13-14

ER -