Out-of-sample embedding of spherical manifold based on constrained least squares

All of the current state-of-the-art nonlinear dimensionality reduction methods attempt to seek the low-dimensional manifold structure by preserving global or local properties of the original data, but without considering the constraint of the manifold structure, thus, there may be a big contrast between the manifold structure result obtained by the nonlinear techniques and the result that we expected. Therefore, it is necessary for us to study the constrained nonlinear dimensionality reduction. In this paper, we study the embedding of a hidden manifold onto a unit sphere by using SMACOF algorithm and propose a method to solve the out-of-sample problem which usually arises in the manifold learning. By converting it into a constrained least squares problem with the spherical structure information, this method avoids reconstructing the neighborhood graph. The application results of 3-D object pose estimation show the effectiveness of our propose method.