  We present a new model which represents data as a mixture of simplices.
Simplices are geometric structures that generalize triangles. We give a simple
geometric understanding that allows us to learn a simplicial structure
efficiently. Our method requires that the data are unit normalized (and thus
lie on the unit sphere). We show that under this restriction, building a model
with simplices amounts to constructing a convex hull inside the sphere whose
boundary facets is close to the data. We call the boundary facets of the convex
hull that are close to the data Activated Simplices. While the total number of
bases used to build the simplices is a parameter of the model, the dimensions
of the individual activated simplices are learned from the data. Simplices can
have different dimensions, which facilitates modeling of inhomogeneous data
sources. The simplicial structure is bounded --- this is appropriate for
modeling data with constraints, such as human elbows can not bend more than 180
degrees. The simplices are easy to interpret and extremes within the data can
be discovered among the vertices. The method provides good reconstruction and
regularization. It supports good nearest neighbor classification and it allows
realistic generative models to be constructed. It achieves state-of-the-art
results on benchmark datasets, including 3D poses and digits.
