  We present a new algorithm for clustering points in R^n. The key property of
the algorithm is that it is affine-invariant, i.e., it produces the same
partition for any affine transformation of the input. It has strong guarantees
when the input is drawn from a mixture model. For a mixture of two arbitrary
Gaussians, the algorithm correctly classifies the sample assuming only that the
two components are separable by a hyperplane, i.e., there exists a halfspace
that contains most of one Gaussian and almost none of the other in probability
mass. This is nearly the best possible, improving known results substantially.
For k > 2 components, the algorithm requires only that there be some
(k-1)-dimensional subspace in which the emoverlap in every direction is small.
Here we define overlap to be the ratio of the following two quantities: 1) the
average squared distance between a point and the mean of its component, and 2)
the average squared distance between a point and the mean of the mixture. The
main result may also be stated in the language of linear discriminant analysis:
if the standard Fisher discriminant is small enough, labels are not needed to
estimate the optimal subspace for projection. Our main tools are isotropic
transformation, spectral projection and a simple reweighting technique. We call
this combination isotropic PCA.
