Tangent space learning and generalization

Manifold learning has attracted considerable attention over the last decade,in which exploring the geometry and topology of the manifold is the central problem.Tangent space is a fundamental tool in discovering the geometry of the manifold.In this paper,we will first review canonical manifold learning techniques and then discuss two fundamental problems in tangent space learning.One is how to estimate the tangent space from random samples,and the other is how to generalize tangent space to ambient space.Previous studies in tangent space learning have mainly focused on how to fit tangent space,and one has to solve a global equation for obtaining the tangent spaces.Unlike these approaches,we introduce a novel method,called persistent tangent space learning(PTSL),which estimates the tangent space at each local neighborhood while ensuring that the tangent spaces vary smoothly on the manifold.Tangent space can be viewed as a point on Grassmann manifold.Inspired from the statistics on Grassmann manifold,we use intrinsic sample total variance to measure the variation of estimated tangent spaces at a single point,and thus,the generalization problem can be solved by estimating the intrinsic sample mean on Grassmann manifold.We validate our methods by various experimental results both on synthetic and real data.