Critical Points to Determine Persistence Homology

In recent years data sets have grown in size and dimension with the proliferation of advanced data acquisition techniques. We have been able to use such data meaningfully not only because the computation power has increased to match the size, but also due to the paradigm shift in data analysis techniques that handle such data. A prime example is Machine Learning (ML). As a result new applications and techniques are emerging more frequently than ever before. Examples include object classification with applications in medicine (e.g., brain image analysis) and security (e.g., face classification) [1]. In many such applications, items in a data set are considered as points in some feature space of the underlying data, enabling us to interpret the data set as a "point cloud" in a suitably identified space. Even though, in certain cases the feature space is easily identifiable, in many other cases identifying a feature space could be a less obvious task.