Pawin Vongmasa, Introduction to Persistent Homology.
In certain studies, researchers aim to understand the nature of data points that they have collected. The technique of Persistent Homology serves as an exploratory step for estimating the shape of data. Related techniques such as clustering were also invented for the same purpose, but they are different in nature. For example, the 0th dimensional persistent homology is similar but slightly inferior to the usual single-linkage clustering, but the concept of persistent homology extends to higher dimensions where other techniques do not extend. Similar to how clusters represent underlying balls where data points are sampled from, the 1st dimensional persistent homology can detect underlying rings, the 2nd dimensional persistent homology can detect underlying closed surfaces (stretched and deformed hollow spheres), and so on. A persistent homology algorithm is a simple extension of the well-known Gaussian elimination. The result of the algorithm is called the persistence barcode, which can be used directly as a visualization or indirectly as an input to other methods for further analysis.
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