Invertible Manifold Learning for Dimension Reduction

It is widely believed that a nonlinear dimension reduction (NLDR) process drops information inevitably in most practical scenarios, and even with the manifold assumption, most existing methods are unable to preserve structure of data after DR due to the loss of information, especially in high-dimensional cases. In the context of manifold learning, we think a good low-dimensional representation should preserve topological and geometric properties of data manifold. To achieve this, the inveribility of a NLDR transformation is required such that the learned representation is reconstructible via its inverse transformation. In this paper, we propose a novel method, called invertible manifold learning (inv-ML), to tackle this problem. A locally isometric smoothness (LIS) constraint for preserving local geometry is applied to a two-stage inv-ML algorithm. Firstly, a homeomorphic sparse coordinate transformation is learned to find the low-dimensional representation without loss of topological information. Secondly, a linear compression is performed on the learned sparse coding, with the trade-off between the target dimension and the incurred information loss. Experiments are conducted on seven datasets, whose results demonstrate that the proposed inv-ML not only achieves better invertible NLDR in comparison with typical existing methods but also reveals the characteristics of the learned manifolds through linear interpolation in latent space. Moreover, we find that the reliability of tangent space approximated by its local neighborhood on real-world datasets is a key to the success of manifold based DR algorithms. The code will be made available soon.