  Analyzing the sub-level sets of the distance to a compact sub-manifold of R d
is a common method in TDA to understand its topology. The distance to measure
(DTM) was introduced by Chazal, Cohen-Steiner and M{\'e}rigot in [7] to face
the non-robustness of the distance to a compact set to noise and outliers. This
function makes possible the inference of the topology of a compact subset of R
d from a noisy cloud of n points lying nearby in the Wasserstein sense. In
practice, these sub-level sets may be computed using approximations of the DTM
such as the q-witnessed distance [10] or other power distance [6]. These
approaches lead eventually to compute the homology of unions of n growing
balls, that might become intractable whenever n is large. To simultaneously
face the two problems of large number of points and noise, we introduce the
k-power distance to measure (k-PDTM). This new approximation of the distance to
measure may be thought of as a k-coreset based approximation of the DTM. Its
sublevel sets consist in union of k-balls, k << n, and this distance is also
proved robust to noise. We assess the quality of this approximation for k
possibly dramatically smaller than n, for instance k = n 1 3 is proved to be
optimal for 2-dimensional shapes. We also provide an algorithm to compute this
k-PDTM.
