Recovering hidden structure from complex and noisy non-linear data is one of the most fundamental problems in machine learning and statistical inference. While such data is often high-dimensional, it is of interest to approximate it with a low-dimensional or even one-dimensional space, since many important aspects of data are often intrinsically low-dimensional. Furthermore, there are many scenarios where the underlying structure is graph-like, e.g, river/road networks or various trajectories. In this paper, we develop a framework to extract, as well as to simplify, a one-dimensional "skeleton" from unorganized data using the Reeb graph. Our algorithm is very simple, does not require complex optimizations and can be easily applied to unorganized high-dimensional data such as point clouds or proximity graphs.

Recovering hidden structure from complex and noisy non-linear data is one of the most fundamental problems in machine learning and statistical inference. While such data is often high-dimensional, it is of interest to approximate it with a low-dimensional or even one-dimensional space, since many important aspects of data are often intrinsically low-dimensional. Furthermore, there are many scenarios where the underlying structure is graph-like, e.g, river/road networks or various trajectories. In this paper, we develop a framework to extract, as well as to simplify, a one-dimensional "skeleton" from unorganized data using the Reeb graph. Our algorithm is very simple, does not require complex optimizations and can be easily applied to unorganized high-dimensional data such as point clouds or proximity graphs. It can also represent arbitrary graph structures in the data.