LLEc

LLEc is a family of extensions to the Local Linear Embedding (LLE) framework that adds extra constraints or curvature-aware terms to improve embeddings in cases of uneven sampling, noise, or nonlinear manifold structure. Like LLE, LLEc aims to preserve local geometry by representing each data point as a weighted combination of its neighbors and then finding a low-dimensional embedding that respects those weights, but with modifications designed to enhance robustness and fidelity on challenging data.

Variants of LLEc differ in the form of the additional terms. In some versions, reconstruction weights are computed under non-negativity and sum-to-one constraints to enforce a convex combination. In others, a curvature penalty is incorporated into the embedding objective to better capture manifolds with varying curvature. The result is often a more stable embedding when data are sparsely sampled or contain outliers, and when the manifold deviates from flat local geometry.

Algorithmic outline. For each point, determine a neighborhood using k-nearest neighbors. Solve a constrained least-squares problem to obtain weights that express the point as a combination of its neighbors, subject to the extra constraints. Compute the low-dimensional embedding by minimizing the weighted reconstruction error sum_i ||y_i − sum_j w_ij y_j||^2, with standard centering and normalization constraints, and with the possible addition of the curvature term.

Applications and limitations. LLEc is used for dimensionality reduction, visualization, and pattern recognition on data with nonlinear manifolds. It can be more robust than vanilla LLE in some scenarios but remains sensitive to the choice of neighborhood size, parameter settings, and computational cost for large datasets.

See also: Local Linear Embedding, Isomap, Laplacian Eigenmaps.