• powerful approach to infer robust qualitative, and sometimes quantitative, information about the structure of data.

• aims at providing well-founded mathematical, statistical and algorithmic methods to infer, analyze and exploit the complex topological and geometric structures underlying the data

• Data often represented as point clouds in Euclidean Space or more general metric spaces

• Points embedded in metric space

• Choice of the metric may be critical to reveal interesting topological and geometric feature of the data.

2. Filtration: Simplicial Complex or a nested family of Simplicial Complex that reflects structure of data at different scales

• Method: Persistent Homology

• Stability to perturbations and noise

Therefore, important to understand the statistical behavior of the inferred features

4. Use extracted information for inference and further use in Machine Learning