Manifold Reconstruction using Deep Residual Network

Abstract

Given a dataset X in Rn, a common assumption is that X is embedded in some unknown smooth compact submanifold in Rn. The goal of manifold reconstruction is to find such manifold from X so that the Hausdorff distance between X and the underlying manifold is small. If additional conditions on X are satisfied, then some geometric properties can be preserved. In [1], a theoret- ical and computational method is proposed to construct a manifold M which approximates the underlying manifold. The key idea in their method is to form local projections which are computed to form tangent spaces of M, then a glu- ing function is applied on X to reconstruct the desired manifold. This method also guarantees the boundedness of serval geometric invariants such as injective radius and sectional curvature. In this study, I will first review the motivation of the above reconstruction algorithm and some properties of the local projec- tion. Second, if local projections are replaced by neural network models, then each local projection can be regarded as a residual block. Therefore, a manifold reconstruction algorithm can be reformulated as a learning process of a residual network. Furthermore, each residual block can be trained individually. Finally, some ongoing works about manifold reconstruction algorithms will be given. ? ?[1] Fefferman, C., Ivanov, S., Kurylev, Y., Lassas, M., Narayanan, H. (2019). ??Reconstruction and Interpolation of Manifolds. I: The Geometric Whitney Problem. Foundations of Computational Mathematics, 1-99.