Information preserving and locally isometric&conformal embedding via Tangent Manifold Learning

In many Data Analysis tasks, one deals with data that are presented in high-dimensional spaces. In practice original high-dimensional data are transformed into lower-dimensional representations (features) preserving certain subject-driven data properties such as distances or geodesic distances, angles, etc. Preserving as much as possible available information contained in the original high-dimensional data is also an important and desirable property of the representation. The real-world high-dimensional data typically lie on or near a certain unknown low-dimensional manifold (Data manifold) embedded in an ambient high-dimensional `observation' space, so in this article we assume this Manifold assumption to be fulfilled. An exact isometric manifold embedding in a low-dimensional space is possible in certain special cases only, so we consider the problem of constructing a `locally isometric and conformal' embedding, which preserves distances and angles between close points. We propose a new geometrically motivated locally isometric and conformal representation method, which employs Tangent Manifold Learning technique consisting in sample-based estimation of tangent spaces to the unknown Data manifold. In numerical experiments, the proposed method compares favourably with popular Manifold Learning methods in terms of isometric and conformal embedding properties as well as of accuracy of Data manifold reconstruction from the sample.