the optimization can be formulated as: After the Gram matrix is learned by semidefinite programming, the output can be obtained via Cholesky decomposition. In particular, the Gram matrix can be written as where is the i-th element of eigenvector of the eigenvalue . It follows that the -th element of the output is . See also Locally linear embedding Isometry (mathematics) Local Tangent Space Alignment Riemannian manifold Energy minimization Notes References Additional material Kilian Q. Weinberger's MVU Matlab code Computational statistics Dimension reduction Optimization algorithms and methods