Constraints on the Neighborhood Size in LLE

Zhengming MA; Jing CHEN; Shuaibin LIAN

doi:10.1587/transinf.E94.D.1636

Constraints on the Neighborhood Size in LLE

Zhengming MA, Jing CHEN, Shuaibin LIAN

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Locally linear embedding (LLE) is a well-known method for nonlinear dimensionality reduction. The mathematical proof and experimental results presented in this paper show that the neighborhood sizes in LLE must be smaller than the dimensions of input data spaces, otherwise LLE would degenerate from a nonlinear method for dimensionality reduction into a linear method for dimensionality reduction. Furthermore, when the neighborhood sizes are larger than the dimensions of input data spaces, the solutions to LLE are not unique. In these cases, the addition of some regularization method is often proposed. The experimental results presented in this paper show that the regularization method is not robust. Too large or too small regularization parameters cannot unwrap S-curve. Although a moderate regularization parameters can unwrap S-curve, the relative distance in the input data will be distorted in unwrapping. Therefore, in order to make LLE play fully its advantage in nonlinear dimensionality reduction and avoid multiple solutions happening, the best way is to make sure that the neighborhood sizes are smaller than the dimensions of input data spaces.

Publication: IEICE TRANSACTIONS on Information Vol.E94-D No.8 pp.1636-1640

Publication Date: 2011/08/01

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Online ISSN: 1745-1361

DOI: 10.1587/transinf.E94.D.1636

Type of Manuscript: PAPER

Category: Pattern Recognition

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Zhengming MA, Jing CHEN, Shuaibin LIAN, "Constraints on the Neighborhood Size in LLE" in IEICE TRANSACTIONS on Information, vol. E94-D, no. 8, pp. 1636-1640, August 2011, doi: 10.1587/transinf.E94.D.1636.
Abstract: Locally linear embedding (LLE) is a well-known method for nonlinear dimensionality reduction. The mathematical proof and experimental results presented in this paper show that the neighborhood sizes in LLE must be smaller than the dimensions of input data spaces, otherwise LLE would degenerate from a nonlinear method for dimensionality reduction into a linear method for dimensionality reduction. Furthermore, when the neighborhood sizes are larger than the dimensions of input data spaces, the solutions to LLE are not unique. In these cases, the addition of some regularization method is often proposed. The experimental results presented in this paper show that the regularization method is not robust. Too large or too small regularization parameters cannot unwrap S-curve. Although a moderate regularization parameters can unwrap S-curve, the relative distance in the input data will be distorted in unwrapping. Therefore, in order to make LLE play fully its advantage in nonlinear dimensionality reduction and avoid multiple solutions happening, the best way is to make sure that the neighborhood sizes are smaller than the dimensions of input data spaces.
URL: https://global.ieice.org/en_transactions/information/10.1587/transinf.E94.D.1636/_p

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@ARTICLE{e94-d_8_1636,
author={Zhengming MA, Jing CHEN, Shuaibin LIAN, },
journal={IEICE TRANSACTIONS on Information},
title={Constraints on the Neighborhood Size in LLE},
year={2011},
volume={E94-D},
number={8},
pages={1636-1640},
abstract={Locally linear embedding (LLE) is a well-known method for nonlinear dimensionality reduction. The mathematical proof and experimental results presented in this paper show that the neighborhood sizes in LLE must be smaller than the dimensions of input data spaces, otherwise LLE would degenerate from a nonlinear method for dimensionality reduction into a linear method for dimensionality reduction. Furthermore, when the neighborhood sizes are larger than the dimensions of input data spaces, the solutions to LLE are not unique. In these cases, the addition of some regularization method is often proposed. The experimental results presented in this paper show that the regularization method is not robust. Too large or too small regularization parameters cannot unwrap S-curve. Although a moderate regularization parameters can unwrap S-curve, the relative distance in the input data will be distorted in unwrapping. Therefore, in order to make LLE play fully its advantage in nonlinear dimensionality reduction and avoid multiple solutions happening, the best way is to make sure that the neighborhood sizes are smaller than the dimensions of input data spaces.},
keywords={},
doi={10.1587/transinf.E94.D.1636},
ISSN={1745-1361},
month={August},}

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TY - JOUR
TI - Constraints on the Neighborhood Size in LLE
T2 - IEICE TRANSACTIONS on Information
SP - 1636
EP - 1640
AU - Zhengming MA
AU - Jing CHEN
AU - Shuaibin LIAN
PY - 2011
DO - 10.1587/transinf.E94.D.1636
JO - IEICE TRANSACTIONS on Information
SN - 1745-1361
VL - E94-D
IS - 8
JA - IEICE TRANSACTIONS on Information
Y1 - August 2011
AB - Locally linear embedding (LLE) is a well-known method for nonlinear dimensionality reduction. The mathematical proof and experimental results presented in this paper show that the neighborhood sizes in LLE must be smaller than the dimensions of input data spaces, otherwise LLE would degenerate from a nonlinear method for dimensionality reduction into a linear method for dimensionality reduction. Furthermore, when the neighborhood sizes are larger than the dimensions of input data spaces, the solutions to LLE are not unique. In these cases, the addition of some regularization method is often proposed. The experimental results presented in this paper show that the regularization method is not robust. Too large or too small regularization parameters cannot unwrap S-curve. Although a moderate regularization parameters can unwrap S-curve, the relative distance in the input data will be distorted in unwrapping. Therefore, in order to make LLE play fully its advantage in nonlinear dimensionality reduction and avoid multiple solutions happening, the best way is to make sure that the neighborhood sizes are smaller than the dimensions of input data spaces.
ER -