Learning to Traverse Image Manifolds UCSD Technical Report

We present a new algorithm, Locally Smooth Manifold Learning (LSML), that learns a warping function from a point on an manifold to its neighbors. Important characteristics of LSML include the ability to recover the structure of the manifold in sparsely populated regions and beyond the support of the provided data. Appli- cations of our proposed technique include embedding with a natural out-of-sample extension and tasks such as tangent distance estimation, frame rate up-conversion, video compression and motion transfer. We show that LSML can recover the structure of the manifold where data is given, and also in regions where it is not, including regions beyond the support of the original data. We propose a number of uses for the recovered warping function W, including embedding with a natural out-of- sample extension, and in the image domain discuss how it can be used for tasks such as computation of tangent distance, image sequence interpolation, compression, and motion transfer. We also show examples where LSML is used to simultaneously learn the structure of multiple "parallel" manifolds, and even generalize to data on new manifolds. Finally, we show that by exploiting the manifold smoothness, LSML is robust under conditions where many embedding methods have difficulty. Related work is presented in Section 2 and the algorithm in Section 3. Experiments on point sets and results on images are shown in Sections 4 and 5, respectively. We conclude in Section 6.