  From a sufficiently large point sample lying on a compact Riemannian
submanifold of Euclidean space, one can construct a simplicial complex which is
homotopy-equivalent to that manifold with high confidence. We describe a
corresponding result for a Lipschitz-continuous function between two such
manifolds. That is, we outline the construction of a simplicial map which
recovers the induced maps on homotopy and homology groups with high confidence
using only finite sampled data from the domain and range, as well as knowledge
of the image of every point sampled from the domain. We provide explicit bounds
on the size of the point samples required for such reconstruction in terms of
intrinsic properties of the domain, the co-domain and the function. This
reconstruction is robust to certain types of bounded sampling and evaluation
noise.
