An introduction to support Vector Machines: and other kernel-based learning methods
An introduction to support Vector Machines: and other kernel-based learning methods
Visualization of a set of parameters characterized by their correlation matrix
Computational Statistics & Data Analysis
Laplacian Eigenmaps for dimensionality reduction and data representation
Neural Computation
Think globally, fit locally: unsupervised learning of low dimensional manifolds
The Journal of Machine Learning Research
Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment
SIAM Journal on Scientific Computing
2005 Special Issue: Cross-entropy embedding of high-dimensional data using the neural gas model
Neural Networks - 2005 Special issue: IJCNN 2005
Building Connected Neighborhood Graphs for Locally Linear Embedding
ICPR '06 Proceedings of the 18th International Conference on Pattern Recognition - Volume 04
Nonlinear Dimensionality Reduction
Nonlinear Dimensionality Reduction
Incremental locally linear embedding algorithm
SCIA'05 Proceedings of the 14th Scandinavian conference on Image Analysis
Interactive visualization and analysis of hierarchical neural projections for data mining
IEEE Transactions on Neural Networks
Optimization and Knowledge-Based Technologies
Informatica
Large-scale multidimensional data visualization: a web service for data mining
ServiceWave'11 Proceedings of the 4th European conference on Towards a service-based internet
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Most of real-life data are not often truly high-dimensional. The data points just lie on a low-dimensional manifold embedded in a high-dimensional space. Nonlinear manifold learning methods automatically discover the low-dimensional nonlinear manifold in a high-dimensional data space and then embed the data points into a low-dimensional embedding space, preserving the underlying structure in the data. In this paper, we have used the locally linear embedding method on purpose to unravel a manifold. In order to quantitatively estimate the topology preservation of a manifold after unfolding it in a low-dimensional space, some quantitative numerical measure must be used. There are lots of different measures of topology preservation. We have investigated three measures: Spearman's rho, Konig's measure (KM), and mean relative rank errors (MRRE). After investigating different manifolds, it turned out that only KM and MRRE gave proper results of manifold topology preservation in all the cases. The main reason is that Spearman's rho considers distances between all the pairs of points from the analysed data set, while KM and MRRE evaluate a limited number of neighbours of each point from the analysed data set.