  A suitable feature representation that can both preserve the data intrinsic
information and reduce data complexity and dimensionality is key to the
performance of machine learning models. Deeply rooted in algebraic topology,
persistent homology (PH) provides a delicate balance between data
simplification and intrinsic structure characterization, and has been applied
to various areas successfully. However, the combination of PH and machine
learning has been hindered greatly by three challenges, namely topological
representation of data, PH-based distance measurements or metrics, and PH-based
feature representation. With the development of topological data analysis,
progresses have been made on all these three problems, but widely scattered in
different literatures. In this paper, we provide a systematical review of PH
and PH-based supervised and unsupervised models from a computational
perspective. Our emphasizes are the recent development of mathematical models
and tools, including PH softwares and PH-based functions, feature
representations, kernels, and similarity models. Essentially, this paper can
work as a roadmap for the practical application of PH-based machine learning
tools. Further, we consider different topological feature representations in
different machine learning models, and investigate their impacts on the protein
secondary structure classification.
