real-world high dimensional data often lie on low-dimensional (sub)manifolds embedded in the high-dimensional space.
Manifold: meaningful low dimensional topological structure of data in high dimensional space
Manifold Learning: discover and model low-dimensional manifolds via learning from data to form a low-dimensional latent embedding representation Recovering/modelling Latent Factors
Rigid vs Non-Rigid Geometry
Rigid shapes are invariant in Euclidean space, while inelastic non-rigid shapes are variant in Euclidean space
Euclidean distance (Extrinsic distance) does not work for non-rigid shapes.
But, there is intrinsic space for inelastic non-rigid shapes (nonlinear manifold) where its intrinsic distance is invariant for any data points in nonlinear manifold.
Manifold learning : Mapping the non-rigid shapes to a latent embedding space where euclidean distance works to preserve the intrinsic distance.
i.e. Model the intrinsic manifold for low-dimensional representation and visualization
Manifold Interpolation
as opposed to linear interpolation
Types
Linear Manifold Learning : Recover linear manifolds Non Linear Manifold Learning : Recover non linear manifolds