  In this paper we revisit the classical method of partitioning classification
and study its convergence rate under relaxed conditions, both for observable
(non-privatised) and for privatised data. Let the feature vector $X$ take
values in $\mathbb{R}^d$ and denote its label by $Y$. Previous results on the
partitioning classifier worked with the strong density assumption, which is
restrictive, as we demonstrate through simple examples. We assume that the
distribution of $X$ is a mixture of an absolutely continuous and a discrete
distribution, such that the absolutely continuous component is concentrated to
a $d_a$ dimensional subspace. Here, we study the problem under much milder
assumptions: in addition to the standard Lipschitz and margin conditions, a
novel characteristic of the absolutely continuous component is introduced, by
which the exact convergence rate of the classification error probability is
calculated, both for the binary and for the multi-label cases. Interestingly,
this rate of convergence depends only on the intrinsic dimension $d_a$.
  The privacy constraints mean that the data $(X_1,Y_1), \dots ,(X_n,Y_n)$
cannot be directly observed, and the classifiers are functions of the
randomised outcome of a suitable local differential privacy mechanism. The
statistician is free to choose the form of this privacy mechanism, and here we
add Laplace distributed noises to the discontinuations of all possible
locations of the feature vector $X_i$ and to its label $Y_i$. Again, tight
upper bounds on the rate of convergence of the classification error probability
are derived, without the strong density assumption, such that this rate depends
on $2\,d_a$.
