Graph induced complex on point data

The efficiency of extracting topological information from point data depends largely on the complex that is built on top of the data points. From a computational viewpoint, the favored complexes for this purpose have so far been Vietoris–Rips and witness complexes. While the Vietoris–Rips complex is simple to compute and is a good vehicle for extracting topology of sampled spaces, its size becomes prohibitively large for reasonable computations. The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structure. We investigate a complex called the graph induced complex that, to some extent, enjoys the advantages of both. It works on a subsample but still retains the power of capturing the topology as the Vietoris–Rips complex. It only needs a graph connecting the original sample points from which it builds a complex on the subsample thus taming the size considerably. We show that, using the graph induced complex one can (i) infer the one dimensional homology of a manifold from a lean subsample, (ii) reconstruct a surface in three dimensions from a sparse subsample without computing Delaunay triangulations, (iii) infer the persistent homology groups of compact sets from a sufficiently dense sample. We provide experimental evidences in support of our theory.