| Experimental data in machine learning is often embedded in a high-dimensional space, which is a set of points sampled from an unknown space. Discovering and u-tilizing the intrinsic topological structure is helpful for the machine learning missions. Topological data analysis focuses on discovering the possible topological structure of the sampled space. Persistent homology is one of the most important technology in topological data analysis. By constructing a filtered complex, persistent homology al-gorithm record and compute the life cycles of some topology invariants. Then it can approximate the possible topological structure of original sample space. So far, there are some theoretical achievements in this field and several applications. However, there is still a big gap between persistent homology and machine learning applications.This paper introduces some theories and algorithms in topological data analysis and we propose prime discriminant simplicial complex (PDSC) by utilizing persistent ho-mology to capture topological structures. It contains the following keys:(1) improved filtered complex construction method, we only extract the structures helpful for our al-gorithm;(2) a barcode technology based on the life cycles of simplices is introduced to select prime discriminant simplicial complex;(3) we classify unlabeled samples based on the nearest projection distances from the samples to the simplicial complexes. We al-so extend the extrapolation ability of these complexes with a projection constraint term.Experiments in simulated and practical datasets indicate that compared with several published algorithms, the proposed PDSC approaches achieve promising performance without losing the structure representation. Meanwhile, due to the advantage of struc-ture representation, there shall be more potential applications which PDSC approaches can be generalized to left for our future research. |
