  Consider a two-class clustering problem where we observe $X_i = \ell_i \mu +
Z_i$, $Z_i \stackrel{iid}{\sim} N(0, I_p)$, $1 \leq i \leq n$. The feature
vector $\mu\in R^p$ is unknown but is presumably sparse. The class labels
$\ell_i\in\{-1, 1\}$ are also unknown and the main interest is to estimate
them.
  We are interested in the statistical limits. In the two-dimensional phase
space calibrating the rarity and strengths of useful features, we find the
precise demarcation for the Region of Impossibility and Region of Possibility.
In the former, useful features are too rare/weak for successful clustering. In
the latter, useful features are strong enough to allow successful clustering.
The results are extended to the case of colored noise using Le Cam's idea on
comparison of experiments.
  We also extend the study on statistical limits for clustering to that for
signal recovery and that for hypothesis testing. We compare the statistical
limits for three problems and expose some interesting insight.
  We propose classical PCA and Important Features PCA (IF-PCA) for clustering.
For a threshold $t > 0$, IF-PCA clusters by applying classical PCA to all
columns of $X$ with an $L^2$-norm larger than $t$. We also propose two
aggregation methods. For any parameter in the Region of Possibility, some of
these methods yield successful clustering. We find an interesting phase
transition for IF-PCA.
  Our results require delicate analysis, especially on post-selection Random
Matrix Theory and on lower bound arguments.
