RATS: Unsupervised manifold learning using low-distortion alignment of tangent spaces

With the ubiquity of high-dimensional datasets in various biological fields, identifying low-dimensional topological manifolds within such datasets may reveal principles connecting latent variables to measurable instances in the world. The reliable discovery of such manifold structure in high-dimensional datasets can prove challenging, however, largely due to the introduction of distortion by leading manifold learning methods. The problem is further exacerbated by the lack of consensus on how to evaluate the quality of the recovered manifolds. Here, we present a novel measure of distortion to evaluate low-dimensional representations obtained using different techniques. We additionally develop a novel bottom-up manifold learning technique called Riemannian Alignment of Tangent Spaces (RATS) that aims to recover low-distortion embeddings of data, including the ability to embed closed manifolds into their intrinsic dimension using a unique tearing process. Compared to previous methods, we show that RATS provides low-distortion embeddings that excel in the visualization and deciphering of latent variables across a range of idealized, biological, and surrogate datasets that mimic real-world data.