Manifold Learning, Near‐isometric Embeddings, Compressed Sensing, Johnson–Lindenstrauss and Some Applications Related to the near Whitney extension problem
Steven B. Damelin · 2024
One of the main challenges in high dimensional data analysis, networks, artificial intelligence, neuroscience, optimal transport and many other related areas of research is dealing with the exponential growth of the computational and sample complexity of several needed generic inference tasks as a function of dimension, a phenomenon termed “the curse of dimensionality”. One intuition that has been put forward to lessen or even obviate the impact of this curse is a manifold hypothesis that the data tends to lie on or near a low dimensional submanifold of the ambient space. It is an interesting problem to study the connections between the manifold hypothesis, the near Whitney extension problem and the optimal transport problem. Classical linear methods for manifold learning include principal component analysis, linear multidimensional scaling and singular value decomposition. Some classical and more recent manifold learning algorithms include Isomap, local linear embedding, Laplacian eigenfunctions, diffusion maps on metric spaces, and so on.