  The probabilistic learning on manifolds (PLoM) introduced in 2016 has solved
difficult supervised problems for the ``small data'' limit where the number N
of points in the training set is small. Many extensions have since been
proposed, making it possible to deal with increasingly complex cases. However,
the performance limit has been observed and explained for applications for
which $N$ is very small (50 for example) and for which the dimension of the
diffusion-map basis is close to $N$. For these cases, we propose a novel
extension based on the introduction of a partition in independent random
vectors. We take advantage of this novel development to present improvements of
the PLoM such as a simplified algorithm for constructing the diffusion-map
basis and a new mathematical result for quantifying the concentration of the
probability measure in terms of a probability upper bound. The analysis of the
efficiency of this novel extension is presented through two applications.
