  Evaluating data separation in a geometrical space is fundamental for pattern
recognition. A plethora of dimensionality reduction (DR) algorithms have been
developed in order to reveal the emergence of geometrical patterns in a low
dimensional visible representation space, in which high-dimensional samples
similarities are approximated by geometrical distances. However, statistical
measures to evaluate directly in the low dimensional geometrical space the
sample group separability attaiend by these DR algorithms are missing.
Certainly, these separability measures could be used both to compare algorithms
performance and to tune algorithms parameters. Here, we propose three
statistical measures (named as PSI-ROC, PSI-PR, and PSI-P) that have origin
from the Projection Separability (PS) rationale introduced in this study, which
is expressly designed to assess group separability of data samples in a
geometrical space. Traditional cluster validity indices (CVIs) might be applied
in this context but they show limitations because they are not specifically
tailored for DR. Our PS measures are compared to six baseline cluster validity
indices, using five non-linear datasets and six different DR algorithms. The
results provide clear evidence that statistical-based measures based on PS
rationale are more accurate than CVIs and can be adopted to control the tuning
of parameter-dependent DR algorithms.
