Persistence in Learning: Persistent Homology and its Applications to Machine Learning

Homology gives a tool to measure the "holes" in topological spaces. Persistent homology extends the theory to define how homology changes over a filtration of a simplicial complex. Further it has been shown that persistent homolgoy is stable under perturbations of the filtration, which motivates its use for studying filtrations derived from data sets. In addition to exploring the theory underlying persistent homology, we explore several methods for building filtered simplicial complexes from data sets including the Rips Complex, the Čech complex, and the Witness complex as well as the relative merits of each approach. We build on these ideas to create a machine learning algorithm based on the persistent homology of a training set. Often in machine learning our training set is not uniformly sampled, but rather sampled from some subset of the feature space which represent sensible input given domain knowledge. This causes issues when trying to classify points which may not be sensible, and thus not accurately represented in any traditional classification model built from the training set. We introduce a classification algorithm called the Nearest Neighbor Persistent Homology which attempts to use the persistent homology of small regions of space to determine if we are in a region of sensible inputs. Testing on real world data, we find that this classifier has higher specificity at the cost of lower sensitivity, so given certain assumptions on the proportion of feature space which is nonsensical, our classifier outperforms the SVM