  This paper considers a canonical clustering problem where one receives
unlabeled samples drawn from a balanced mixture of two elliptical distributions
and aims for a classifier to estimate the labels. Many popular methods
including PCA and k-means require individual components of the mixture to be
somewhat spherical, and perform poorly when they are stretched. To overcome
this issue, we propose a non-convex program seeking for an affine transform to
turn the data into a one-dimensional point cloud concentrating around $-1$ and
$1$, after which clustering becomes easy. Our theoretical contributions are
two-fold: (1) we show that the non-convex loss function exhibits desirable
geometric properties when the sample size exceeds some constant multiple of the
dimension, and (2) we leverage this to prove that an efficient first-order
algorithm achieves near-optimal statistical precision without good
initialization. We also propose a general methodology for clustering with
flexible choices of feature transforms and loss objectives.
