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Orbit Learning using Convex Optimization

Recently, learning approaches have been brought to bear on nonlinear datasets by assuming samples lie on a low-dimensional Riemannian manifold in the embedding space. One solution is to model local variations linearly [1]. A fundamental difficulty with such solutions is they cannot extrapolate or generalize far from the training data when the manifold is curved. Consequently, the manifold needs to be densely sampled; particularly at high curvature regions. To deal with this serious problem, we consider manifolds that emerge from group actions on the data as in [2]: the manifold is actually an orbit of unknown transformations. For instance, images of a 3D object from varying viewpoints can be seen as the result of multiple nonlinear operators that act upon a small set of prototype views. Although this may seem like a non convex optimization problem, we propose a formulation that is convex and produces global estimates of the group generators while faithfully approximating data with a low-dimensional orbit. This approach allows us to generalize to new parts of the space and to extrapolate the manifolds nonlinearly while still keeping a compact and regularized description of the data.