We generalize the local-feature size definition of adaptive sampling used in surface reconstruction to relate it to an alternative metric on Euclidean space. In the new metric, adaptive samples become uniform samples, making it simpler both to give adaptive sampling versions of homological inference results and to prove topological guarantees using the critical points theory of distance functions. This ultimately leads to an algorithm for homology inference from samples whose spacing depends on their distance to a discrete representation of the complement space.

@article{cavanna20adaptive,
  title = {Adaptive Metrics for Adaptive Samples},
  author = {Nicholas J. Cavanna and Donald R. Sheehy},
  journal = {Algorithms},
  volume = {13},
  number = {8},
  pages = {200:1--15},
  year = {2020}}