  Machine learning and data analysis now finds both scientific and industrial
application in biology, chemistry, geology, medicine, and physics. These
applications rely on large quantities of data gathered from automated sensors
and user input. Furthermore, the dimensionality of many datasets is extreme:
more details are being gathered about single user interactions or sensor
readings. All of these applications encounter problems with a common theme: use
observed data to make inferences about the world. Our work obtains the first
provably efficient algorithms for Independent Component Analysis (ICA) in the
presence of heavy-tailed data. The main tool in this result is the centroid
body (a well-known topic in convex geometry), along with optimization and
random walks for sampling from a convex body. This is the first algorithmic use
of the centroid body and it is of independent theoretical interest, since it
effectively replaces the estimation of covariance from samples, and is more
generally accessible.
  This reduction relies on a non-linear transformation of samples from such an
intersection of halfspaces (i.e. a simplex) to samples which are approximately
from a linearly transformed product distribution. Through this transformation
of samples, which can be done efficiently, one can then use an ICA algorithm to
recover the vertices of the intersection of halfspaces.
  Finally, we again use ICA as an algorithmic primitive to construct an
efficient solution to the widely-studied problem of learning the parameters of
a Gaussian mixture model. Our algorithm again transforms samples from a
Gaussian mixture model into samples which fit into the ICA model and, when
processed by an ICA algorithm, result in recovery of the mixture parameters.
Our algorithm is effective even when the number of Gaussians in the mixture
grows polynomially with the ambient dimension
