  Topological data analysis is a recent and fast growing field that approaches
the analysis of datasets using techniques from (algebraic) topology. Its main
tool, persistent homology (PH), has seen a notable increase in applications in
the last decade. Often cited as the most favourable property of PH and the main
reason for practical success are the stability theorems that give theoretical
results about noise robustness, since real data is typically contaminated with
noise or measurement errors. However, little attention has been paid to what
these stability theorems mean in practice. To gain some insight into this
question, we evaluate the noise robustness of PH on the MNIST dataset of
greyscale images. More precisely, we investigate to what extent PH changes
under typical forms of image noise, and quantify the loss of performance in
classifying the MNIST handwritten digits when noise is added to the data. The
results show that the sensitivity to noise of PH is influenced by the choice of
filtrations and persistence signatures (respectively the input and output of
PH), and in particular, that PH features are often not robust to noise in a
classification task.
