Spectral Simplicial Theory for Feature Selection and Applications to Genomics

The utility of the Čech complex is ensured by the nerve theorem: if the elements of were sampled from a topological space M, under certain sampling assumptions the Čech complex approximates the topology of M. In particular, the 1-skeleton (i.e. the set of 0-and 1-simplices) of the Čech complex is a graph of the type considered in manifold learning approaches. Since computing triple and higher-order intersections among balls is computationally costly, the Vietoris-Rips complex (defined as the clique complex of the 1-skeleton of a Čech complex) is often used in practice as an approximation to the Čech complex (6). Additionally, Uniform Manifold Approximation and Projection (UMAP) (7) and Mapper (8) can be used to generate reduced abstract simplicial complex representations of the data. Figure 2. Čech complex constructed from intersections of fixed-radius balls centered at the elements of the point cloud . In this example, the elements of are ordered according to the horizontal coordinate.