Detecting Intrinsic Loops Underlying Data Manifold

Abstract

Detecting intrinsic loop structures of a data manifold is the necessary prestep for the proper employment of the manifold learning techniques and of fundamental importance in the discovery of the essential representational features underlying the data lying on the loopy manifold. An effective strategy is proposed to solve this problem in this study. In line with our intuition, a formal definition of a loop residing on a manifold is first given. Based on this definition, theoretical properties of loopy manifolds are rigorously derived. In particular, a necessary and sufficient condition for detecting essential loops of a manifold is derived. An effective algorithm for loop detection is then constructed. The soundness of the proposed theory and algorithm is validated by a series of experiments performed on synthetic and real-life data sets. In each of the experiments, the essential loops underlying the data manifold can be properly detected, and the intrinsic representational features of the data manifold can be revealed along the loop structure so detected. Particularly, some of these features can hardly be discovered by the conventional manifold learning methods.