Tangent space estimation for smooth embeddings of Riemannian manifolds

Numerous dimensionality reduction problems in data analysis involve the recovery of low-dimensional models or the learning of manifolds underlying sets of data. Many manifold learning methods require the estimation of the tangent space of the manifold at a point from locally available data samples. Local sampling conditions such as (i) the size of the neighborhood (sampling width) and (ii) the number of samples in the neighborhood (sampling density) affect the performance of learning algorithms. In this work, we propose a theoretical analysis of local sampling conditions for the estimation of the tangent space at a point P lying on a m-dimensional Riemannian manifold S in R^n. Assuming a smooth embedding of S in R^n, we estimate the tangent space T_P S by performing a Principal Component Analysis (PCA) on points sampled from the neighborhood of P on S. Our analysis explicitly takes into account the second order properties of the manifold at P, namely the principal curvatures as well as the higher order terms. We consider a random sampling framework and leverage recent results from random matrix theory to derive conditions on the sampling width and the local sampling density for an accurate estimation of tangent subspaces. We measure the estimation accuracy by the angle between the estimated tangent space and the true tangent space T_P S and we give conditions for this angle to be bounded with high probability. In particular, we observe that the local sampling conditions are highly dependent on the correlation between the components in the second-order local approximation of the manifold. We finally provide numerical simulations to validate our theoretical findings

Local and global regressive mapping for manifold learning with out-of-sample extrapolation

Over the past few years, a large family of manifold learning algorithms have been proposed, and applied to various applications. While designing new manifold learning algorithms has attracted much research attention, fewer research efforts have been focused on out-ofsample extrapolation of learned manifold. In this paper, we propose a novel algorithm of manifold learning. The proposed algorithm, namely Local and Global Regressive Mapping (LGRM), employs local regression models to grasp the manifold structure. We additionally impose a global regression term as regularization to learn a model for out-of-sample data extrapolation. Based on the algorithm, we propose a new manifold learning framework. Our framework can be applied to any manifold learning algorithms to simultaneously learn the low dimensional embedding of the training data and a model which provides explicit mapping of the outof- sample data to the learned manifold. Experiments demonstrate that the proposed framework uncover the manifold structure precisely and can be freely applied to unseen data