Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment
SIAM Journal on Scientific Computing
Building k Edge-Disjoint Spanning Trees of Minimum Total Length for Isometric Data Embedding
IEEE Transactions on Pattern Analysis and Machine Intelligence
Selection of the optimal parameter value for the Isomap algorithm
Pattern Recognition Letters
Local relative transformation with application to isometric embedding
Pattern Recognition Letters
Using graph algebra to optimize neighborhood for isometric mapping
IJCAI'07 Proceedings of the 20th international joint conference on Artifical intelligence
Parameterless isomap with adaptive neighborhood selection
DAGM'06 Proceedings of the 28th conference on Pattern Recognition
Supervised nonlinear dimensionality reduction for visualization and classification
IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics
Isometric sliced inverse regression for nonlinear manifold learning
Statistics and Computing
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The construction of the neighborhood is a critical problem of manifold learning. Most of manifold learning algorithms use a stable neighborhood parameter (such as k-NN), but it may not work well for the entire manifold, since manifold curvature and sampling density may vary over the manifold. Although some dynamical neighborhood algorithms have been proposed, they are limited by either another global parameter or an assumption. This paper proposes a new approach to select the dynamical neighborhood for each point while constructing the tangent subspace based on the sampling density and the manifold curvature. And the parameters of the approach can be automatically determined by computing the correlation coefficient of the matrices of geodesic distances between pairs of points in input and output spaces. When we apply it to ISOMAP, the results of experiments on the synthetic data as well as the real world patterns demonstrate that the proposed approach can efficiently maintain an accurate low dimensional representation of the manifold data with less distortion, and give higher average classification rate compared to others.