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 . 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. Specifically, we consider the case when the data points lie on a manifold M of dimension d, which is embedded in the higher-dimensional input space with dimension D. A method is called manifold-adaptive if its sample complexity can be bounded by a quantity whose exponent depends only on d and not on D. Thus a manifold-adaptive method may enjoy a considerably better sample complexity whenever d is much smaller than D. Although there are many learning methods that are designed to be manifold adaptive (or manifold friendly), they more often than not lack a rigorous proof of this property (one exception is the recent work of Scott and Nowak on dyadic decision trees in a classification context, cf. ). The first method, proposed by us earlier in , concerns the problem of estimating the dimension of a manifold based on points sampled from it. The second method is the classical k-nearest neighbor regressor. We find it intriguing that this method was not specifically designed to be manifold adaptivity, yet it is relatively simple to prove that it possesses this property.