localitypreserving

Locality-preserving refers to a principle in data analysis and machine learning where the primary aim of a transformation or representation is to preserve the local structure of the data—i.e., the neighborhood relations among data points—while possibly preserving limited global geometry.

In dimensionality reduction, locality-preserving methods attempt to map high-dimensional data to a lower-dimensional space such that points that are close in the original space remain close in the embedding. Common approaches include Locality Preserving Projections (LPP) and Laplacian Eigenmaps. These methods typically build a weighted graph over the data, using k-nearest neighbors or an epsilon neighborhood to define adjacency, and a weight matrix W that encodes similarity between neighboring points. An objective function often takes the form of minimizing the weighted sum of squared distances in the embedding: sum_{i,j} W_{ij} ||y_i - y_j||^2 subject to a normalization constraint. This leads to a generalized eigenvalue problem that yields the embedding coordinates from the smallest nonzero eigenvectors of a graph Laplacian L = D - W, where D is the degree matrix.

Locality-preserving properties are advantageous when data lie on a low-dimensional manifold within a high-dimensional space, but they can be sensitive to the choice of neighborhood, graph construction, and noise. They may also distort global relationships and often require methods to extend the embedding to new data points (out-of-sample extension). Beyond dimensionality reduction, locality-preserving ideas appear in locality-preserving hashing and other similarity-preserving techniques that aim to keep nearby items close in the transformed representation.