A New View of Classification in Astronomy with the Archetype Technique: An Astronomical Case of the NP-complete Set Cover Problem

We introduce a new generic Archetype technique for source classification and identification, based on the NP-complete set cover problem (SCP) in computer science and operations research (OR). We have developed a new heuristic SCP solver, by combining the greedy algorithm and the Lagrangian Relaxation (LR) approximation method. We test the performance of our code on the test cases from Beasley's OR Library and show that our SCP solver can efficiently yield solutions that are on average 99% optimal in terms of the cost. We discuss how to adopt SCP for classification purposes and put forward a new Archetype technique. We use an optical spectroscopic dataset of extragalactic sources from the Sloan Digital Sky Survey (SDSS) as an example to illustrate the steps of the technique. We show how the technique naturally selects a basis set of physically-motivated archetypal systems to represent all the extragalactic sources in the sample. We discuss several key aspects in the technique and in any general classification scheme, including distance metric, dimensionality, and measurement uncertainties. We briefly discuss the relationships between the Archetype technique and other machine-learning techniques, such as the $k$-means clustering method. Finally, our code is publicly available and the technique is generic and easy to use and expand. We expect that it can help maximize the potential for astrophysical sciences of the low-S/N spectroscopic data from future dark-energy surveys, and can find applications in many fields of astronomy, including the formation and evolution of a variety of astrophysical systems, such as galaxies, stars and planets.