Local Sampling Analysis for Quadratic Embeddings of Riemannian Manifolds

In this work we do a theoretical analysis of the local sampling conditions for points lying on a quadratic embedding of a Riemannian manifold in a Euclidean space. The embedding is assumed to be quadratic at a reference point $P$. Our analysis is based on the following criteria: (i) Local reconstruction error (ii) Local tangent space estimation accuracy. In the local reconstruction error analysis we describe sampling conditions in the neighbourhood of $P$ such that the average reconstruction error of the samples after orthogonal projection on the local tangent space, satisfies a given upper bound. We derive a lower bound on the number of neighbouring samples which probabilistically guarantees that a predefined local reconstruction error criterion will be satisfied. In local tangent space estimation analysis, we analyze the locally estimated linear subspace, which is optimal in the least squares sense and passes through $P$. The tangent space at $P$ is estimated using the samples lying in its neighbourhood. Sampling conditions for the neighbourhood points are derived so that the angle'' between the estimated tangent space and the original tangent space at $P$ is upper bounded. We again consider both probabilistic and non-probabilistic sampling conditions for this criterion. We derive a lower bound on the number of neighbouring samples which probabilistically guarantees an upper bound on the angle'' between the estimated tangent space and the original tangent space.