Fixed point laplacian mapping: a geometrically correct manifold learning algorithm

Dimensionality reduction (DR) and manifold learning (ManL) have been applied extensively in many machine learning tasks, including computer vision, image analysis and pattern recognition just to name a few. However, the geometrical correctness of DR and ManL models learning results is largely neglected. In this work, we investigate this important aspect of some widely used DR and ManL methods through the lens of the chart map function, which is the essential part of the definition of a manifold. It turns out that the mapping functions induced by these methods do not have the injectivity (i.e. one-to-one mapping) between ambient space and latent space. This poses the distinguishability problem for down-stream tasks. Without injectivity, two distinct points on a manifold may be projected to the same point in the latent space learnt by DR or ManL methods, and hence the later process cannot separate them apart, resulting in inevitable errors. To address this problem, we provide a provably correct algorithm called fixed points Laplacian mapping (FPLM), which has the geometric guarantee to find a representation of a manifold with injectivity by using a simplical complex generated on manifold. We further discuss the property of our proposed method via various aspects, including its link to the popular graph neural networks (GNNs) and deep neural networks (DNNs). Its geometric correctness is demonstrated by extensive experimental results and theoretical proofs. Moreover, experiments also show our method is capable of evaluating the quality of simplex decomposition of the manifold and detecting manifold intrinsic dimensions for real-world datasets.