  We present a signal representation framework called the sparse manifold
transform that combines key ideas from sparse coding, manifold learning, and
slow feature analysis. It turns non-linear transformations in the primary
sensory signal space into linear interpolations in a representational embedding
space while maintaining approximate invertibility. The sparse manifold
transform is an unsupervised and generative framework that explicitly and
simultaneously models the sparse discreteness and low-dimensional manifold
structure found in natural scenes. When stacked, it also models hierarchical
composition. We provide a theoretical description of the transform and
demonstrate properties of the learned representation on both synthetic data and
natural videos.
