Confidence Regions for Barcodes

The results of [3] suggest a way to sort which barcode intervals correctly belong to the underlying manifold, M , and which can be regarded as topological noise. In particular, when barcodes are computed using sublevel sets of their distance function (see eq 12), all of the barcodes that start later than a linear function of W2(F̂n, F ) can be assumed to correctly reflect the betti numbers of M , where F is the distribution of points on M , possibly with convoluted noise, and F̂n is the corresponding empirical distribution. In practice, since the underlying distribution F is unknown, W2(F̂n, F ) cannot be computed. But, since it is true that W2 metrizes weak convergence, and F̂n d −→ F , we can hope that it has a limiting distribution independent of F . Then one could define a confidence region for the bounds suggested by [3], and any barcode intervals starting early and persisting long enough to escape these bounds can be assumed corresponding to true betti numbers at some confidence level α. See figure 1 for an example.