Topological data analysis (TDA) is a growing branch of mathematics that concerns the study of the shape of inherently high-dimensional data. One particularly useful tool is that of persistent homology, where n-dimensional topological features of the data are encoded into a collection of coordinates in the plane, called a persistence diagram. Recently, persistence diagrams have demonstrated their usefulness in generating summary statistics for machine learning algorithms. However, for particularly large data sets or noisy images, approximations or smoothing must first be applied to either make computations possible or to clean the signature of the features. In this talk, we introduce persistent homology and show a recent result that makes it possible to compute rigorous bounds on the amount of error these approximations introduce into the system being studied. We will focus on two examples: estimating the shape of a

large point cloud and studying features in noisy images.