Isometric Manifold Learning Using Hierarchical Flow

Abstract

We propose the Hierarchical Flow (HF) model constrained by isometric regularizations for manifold learning that combines manifold learning goals such as dimensionality reduction, inference, sampling, projection and density estimation into one unified framework. Our proposed HF model is regularized to not only produce embeddings preserving the geometric structure of the manifold, but also project samples onto the manifold in a manner conforming to the rigorous definition of projection. Theoretical guarantees are provided for our HF model to satisfy the two desired properties. In order to detect the real dimensionality of the manifold, we also propose a two-stage dimensionality reduction algorithm, which is a time-efficient algorithm thanks to the hierarchical architecture design of our HF model. Experimental results justify our theoretical analysis, demonstrate the superiority of our dimensionality reduction algorithm in cost of training time, and verify the effect of the aforementioned properties in improving performances on downstream tasks such as anomaly detection.