  In this paper, we present a method for denoising and reconstruction of
low-dimensional manifold in high-dimensional space. We suggest a
multidimensional extension of the Locally Optimal Projection algorithm which
was introduced by Lipman et al. in 2007 for surface reconstruction in 3D. The
method bypasses the curse of dimensionality and avoids the need for carrying
out dimensional reduction. It is based on a non-convex optimization problem,
which leverages a generalization of the outlier robust L1-median to higher
dimensions while generating noise-free quasi-uniformly distributed points
reconstructing the unknown low-dimensional manifold. We develop a new algorithm
and prove that it converges to a local stationary solution with a bounded
linear rate of convergence in case the starting point is close enough to the
local minimum. In addition, we show that its approximation order is $O(h^2)$,
where $h$ is the representative distance between the given points. We
demonstrate the effectiveness of our approach by considering different manifold
topologies with various amounts of noise, including a case of a manifold of
different co-dimensions at different locations.
