  A novel topological-data-analytical (TDA) method is proposed to distinguish,
from noise, small holes surrounded by high-density regions of a probability
density function. The proposed method is robust against additive noise and
outliers. Traditional TDA tools, like those based on the distance filtration,
often struggle to distinguish small features from noise, because both have
short persistences. An alternative filtration, called the Robust Density-Aware
Distance (RDAD) filtration, is proposed to prolong the persistences of small
holes of high-density regions. This is achieved by weighting the distance
function by the density in the sense of Bell et al. The concept of
distance-to-measure is incorporated to enhance stability and mitigate noise.
The persistence-prolonging property and robustness of the proposed filtration
are rigorously established, and numerical experiments are presented to
demonstrate the proposed filtration's utility in identifying small holes.
