Systems and methods perform Laplacian Principal Components Analysis (LPCA). In one implementation, an exemplary system receives multidimensional data and reduces dimensionality of the data by locally optimizing a scatter of each local sample of the data. The optimization includes summing weighted distances between low dimensional representations of the data and a mean. The weights of the distances can be determined by a coding length of each local data sample. The system can globally align the locally optimized weighted scatters of the local samples and provide a global projection matrix. The LPCA improves performance of such applications as face recognition and manifold learning.