  The stability of persistent homology has led to wide applications of the
persistence diagram as a trusted topological descriptor in the presence of
noise. However, with the increasing demand for high-dimension and
low-sample-size data processing in modern science, it is questionable whether
persistence diagrams retain their reliability in the presence of
high-dimensional noise. This work aims to study the reliability of persistence
diagrams in the high-dimension low-sample-size data setting. By analyzing the
asymptotic behavior of persistence diagrams for high-dimensional random data,
we show that persistence diagrams are no longer reliable descriptors of
low-sample-size data under high-dimensional noise perturbations. We refer to
this loss of reliability of persistence diagrams in such data settings as the
curse of dimensionality on persistence diagrams. Next, we investigate the
possibility of using normalized principal component analysis as a method for
reducing the dimensionality of the high-dimensional observed data to resolve
the curse of dimensionality. We show that this method can mitigate the curse of
dimensionality on persistence diagrams. Our results shed some new light on the
challenges of processing high-dimension low-sample-size data by persistence
diagrams and provide a starting point for future research in this area.
