LDLE: Low Distortion Local Eigenmaps
This addresses a challenge in manifold learning for handling complex topological structures, offering a novel approach with potential applications in data analysis, though it appears incremental in advancing existing embedding techniques.
The paper tackles the problem of embedding closed and non-orientable manifolds into their intrinsic dimension by introducing LDLE, a technique that constructs low-distortion local views and registers them globally, resulting in preserved distances up to a constant scale compared to higher distortion from other methods.
We present Low Distortion Local Eigenmaps (LDLE), a manifold learning technique which constructs a set of low distortion local views of a dataset in lower dimension and registers them to obtain a global embedding. The local views are constructed using the global eigenvectors of the graph Laplacian and are registered using Procrustes analysis. The choice of these eigenvectors may vary across the regions. In contrast to existing techniques, LDLE can embed closed and non-orientable manifolds into their intrinsic dimension by tearing them apart. It also provides gluing instruction on the boundary of the torn embedding to help identify the topology of the original manifold. Our experimental results will show that LDLE largely preserved distances up to a constant scale while other techniques produced higher distortion. We also demonstrate that LDLE produces high quality embeddings even when the data is noisy or sparse.