Multidimensional persistent homology is stable

@article{Cerri2009MultidimensionalPH,
  title={Multidimensional persistent homology is stable},
  author={Andrea Cerri and Barbara Di Fabio and Massimo Ferri and Patrizio Frosini and Claudia Landi},
  journal={arXiv: Algebraic Topology},
  year={2009}
}
Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with re- spect to function perturbations. More precisely, we construct a distance be- tween rank invariants such that small changes of the function imply only small changes of the rank invariant. This result… 

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