Figure 2

A manifold ℳ can be viewed as a nonlinear surface in ℝ ^N. When the mapping between θ and x _θ is well-behaved, as we trace out a path in the parameter space Θ, we trace out a similar path on ℳ. A random projection Φ from ℝ^N to a lower dimensional space ℝ^M can provide a stable embedding of ℳ, preserving all pairwise distances, and therefore preserving the structure within an ensemble of images. The goal of a manifold lifting algorithm is to recover an ensemble of images from their low-dimensional measurements.