Toward Manifold-Adaptive Learning

Inputs coming from high-dimensional spaces are common in many real-world problems such as a robot control with visual inputs. Yet learning in such cases is in general difficult, a fact often referred to as the “curse of dimensionality”. In particular, in regression or classification, in order to achieve a certain accuracy algorithms are known to require exponentially many samples in the dimension of the inputs in the worst-case [1]. The exponential dependence on the input dimension forces us to develop methods that are efficient in exploiting regularities of the data. Classically, smoothness is the best known example of such a regularity. In this abstract we outline two methods for two problems that are efficient in exploiting when the data points lie on a low dimensional submanifold of the input space.