 
Summary: Toward ManifoldAdaptive Learning
Amir massoud Farahmand Csaba Szepesv´ari
Department of Computing Science, University of Alberta
Edmonton, Canada
{amir, szepesva}@cs.ualberta.ca
JeanYves Audibert
CERTIS  Ecole des Ponts
19, rue Alfred Nobel  Cit´e Descartes, 77455 MarnelaVall´ee France
audibert@cermics.enpc.fr
1 Introduction
Inputs coming from highdimensional spaces are common in many realworld 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 worstcase [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.
Specifically, we consider the case when the data points lie on a manifold M of dimension d, which
