Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis

Understand

We consider the problem of recovering a $d-$dimensional manifold $\mathcal{M} \subset \mathbb{R}^n$ when provided with noiseless samples from $\mathcal{M}$.

• There are many algorithms (e.g., Isomap) that are used in practice to fit manifolds and thus reduce the dimensionality of a given data set.
• Ideally, the estimate $\mathcal{M}_\mathrm{put}$ of $\mathcal{M}$ should be an actual manifold of a certain smoothness; furthermore, $\mathcal{M}_\mathrm{put}$ should be arbitrarily close to $\mathcal{M}$ in Hausdorff distance given a large enough sample.
• Generally speaking, existing manifold learning algorithms do not meet these criteria.