LPPbased

LPPbased refers to methods that use Locality Preserving Projections (LPP) as a core component for dimensionality reduction, feature extraction, or data representation. LPP-based approaches aim to map high-dimensional data to a lower-dimensional space while preserving local neighborhood relationships, thereby maintaining the intrinsic geometry of the data.

Locality Preserving Projections was introduced by He and Niyogi in 2004 as a linear alternative to nonlinear manifold learning methods like Laplacian eigenmaps. The method constructs a nearest-neighbor graph on the data, assigns weights W_ij to reflect pairwise similarities, computes the degree matrix D and the Laplacian L = D − W, and seeks a projection matrix A such that the projected coordinates y_i = A^T x_i minimize a cost that preserves locality. In practice, one solves a generalized eigenvalue problem X L X^T A = X D X^T A Λ, selecting the eigenvectors corresponding to the smallest eigenvalues to form A. The reduced representation of a sample is y = A^T x.

LPP-based methods extend this framework in various ways. Kernel LPP (KLPP) applies a kernel trick to capture nonlinear structure. Supervised LPP uses label information to emphasize within-class similarity and between-class separability. Other variants include robust, semi-supervised, and discriminant LPP approaches.

Applications of LPP-based methods span image and video analysis, face recognition, handwriting and texture classification, and general pattern recognition tasks where preserving local geometry improves downstream learning. They are often used as a pre-processing or feature extraction step before classifiers or clustering algorithms.

Advantages include preserving local structure with a linear mapping and relatively efficient computation. Limitations involve dependence on graph construction, sensitivity to parameter choices, and potential failure to capture global structure or highly nonlinear manifolds. See also Laplacian eigenmaps and PCA. References: He and Niyogi, Locality Preserving Projections, 2004.