Manifold alignment using discrete surface Ricci flow

Manifold alignment is useful to extract the shared latent structure among multiple data sets and the similarity among different datasets. As many kinds of real world data can be analyzed using low dimensional representations, manifold alignment algorithms can be used in a wide range of applications, such as data mining. In this paper, we propose a three-stage approach to manifold alignment using discrete surface Ricci flow. Our approach transforms the original intrinsic manifolds to hyper spheres using conformal mapping in the first stage, and then zooms these hyper spheres into the same scale and aligns them in the following stages. We describe in details about our algorithm, its theoretical principles, our experimental results, and the comparison to previous alignment methods. To prove the effectiveness of our algorithm, three kinds of experiments are presented, including a toy dataset, one containing parallel corpus of parliament proceedings and another containing both images and texts. With these experiments, the latent utility in discovering the similarity among different kinds of data sets can be demonstrated, whether within the same kind of data or across different kinds of modals of data.