Besides, some manifold learning methods have been proposed including Laplacian eigenmaps [19, 20],
locally linear embedding [21], and Isometric Mapping method [22, 23].
The traditional feature extraction methods such as principal component analysis (PCA) cannot obtain the local structure of the samples, and
locally linear embedding (LLE) cannot obtain the global structure of the samples.
Locally Linear Embedding. Local linear embedding (LLE) is a nonlinear dimensionality reduction method widely used these years.
The graph learning based manifold learning methods [9-15], such as LE (Laplacian Eigenmaps) or LPP (Locality Preserving Projections) [9-10], LLE (
Locally Linear Embedding) [11], DM (Diffusion Maps) [12], Isomap [13] and LDE (Locally Discriminant Embedding) or MFA (Marginal Fisher Analysis) [14-15] can be represented as the unified graph embedding framework [15], least-squares framework [16] or their extensive forms.
To address this problem, some other algorithms have been developed, such as
Locally Linear Embedding (LLE) [3, 4], Isomap [5], Laplacian Eigenmaps [6], Locality Preserving Projections [7, 8], Neighborhood Preserving Embedding [9], and Tangent Distance Preserving Mapping [10].
In 2000, two novel methods for manifold learning, isometric feature map method (Isomap) [5] and the
locally linear embedding method (LLE) [6], have drawn great interests.
LPP is to find the optimal linear approximations to the eigen functions of the Laplace Beltrami operator on the manifold, sharing many of the data representation properties of nonlinear dimensionality reduction such as Laplacian Eigen maps or
Locally Linear Embedding. LPP keeps nearby data pairs in the original space close in the embedding space, by which nonlinear data can be embedded without losing its local structure.
The representative algorithms include
locally linear embedding (LLE), Isomap, and Laplacian eigenmap [5].