years a lot of work on unsupervised learn- ing based on characterizing a possibly non-linear manifold near which the data would lie, such as Locally Linear Embedding (LLE) (Roweis and Saul, 2000), Isomap ...

Canonical Decomposition This chapter begins with the ?rst general result on 3 manifolds, Kneser’s theorem that every compact orientable 3 manifold M decomposes uniquely as a connected sum ...