Iterative projections for signal identification on manifolds: Global recovery guarantees

We introduce an algorithm known as Manifold Iterative Projection to solve the problem of recovering an unknown high-dimensional signal contained in a low-dimensional sub-manifold from a few linear measurements. The algorithm provably and robustly recovers any unknown signal on the manifold, provided the measurement operator is benign with respect to the manifold. A variant of the algorithm provably tracks slowly time-varying signals on the manifold. Our results are intimately related to, and indeed rely on, the existence of stable embeddings of manifolds via linear maps.