Nonparametric regressors on ${R^D}$ are subject to a so-called 'curse of dimension':
there always exists a distribution such that the regressor, given $n$ samples,
converges at 
a rate of the form $n^{-2/(2+D)}$. In other words, we need a sample size $n$
exponential in $D$ in order to approximate the unknown function. Fortunately, as
we will see, there are common real-world scenarios where better regression rates
are achievable for data in ${R^D}$. 

We derive notions of 'local dimension' that better capture real-world
scenarios. The local dimension is often smaller than the dimension $D$. We show
that common 'local regressors' 
operating on ${R^D}$, converge at rates that depend just on the local dimension and
not on $D$. These are often better rates than the worst-case of $n^{-2/(2+D)}$.