  We consider a problem of great practical interest: the repairing and recovery
of a low-dimensional manifold embedded in high-dimensional space from noisy
scattered data. Suppose that we observe a point cloud sampled from the
low-dimensional manifold, with noise, and let us assume that there are holes in
the data. Can we recover missing information inside the holes? While in
low-dimension the problem was extensively studied, manifold repairing in high
dimension is still an open problem. We introduce a new approach, called
Repairing Manifold Locally Optimal Projection (R-MLOP), that expands the MLOP
method introduced by Faigenbaum-Golovin et al. in 2020, to cope with manifold
repairing in low and high-dimensional cases. The proposed method can deal with
multiple holes in a manifold. We prove the validity of the proposed method, and
demonstrate the effectiveness of our approach by considering different manifold
topologies, for single and multiple holes repairing, in low and high
dimensions.
