Geometric Manifold Approximation using Union of Tangent Patches

This paper addresses the problem of data-adaptive learning of the ambient geometry of a nonlinear, non-intersecting submanifold of a Euclidean space. It accomplishes this goal by exploiting the local linearity of the (sub)manifold and approximating it using a union of tangent patches (UoTP). In addition, it translates the problem of projecting a new data point onto the learned UoTP into a series of convex optimization problems. It then derives a procedure for encoding (projecting) data points onto a UoTP that involves an efficient solution to each of the posed optimization problems. Finally, it demonstrates the value of capturing the geometry of manifolds by comparing the superior denoising performance of the proposed framework on both synthetic and real data sampled from nonlinear manifolds with that of stat-of-the-art denoising algorithms.