  In this paper we show that a large class of Latent variable models, such as
Mixed Membership Stochastic Block(MMSB) Models, Topic Models, and Adversarial
Clustering, can be unified through a geometric perspective, replacing model
specific assumptions and algorithms for individual models. The geometric
perspective leads to the formulation: \emph{find a latent $k-$ polytope $K$ in
${\bf R}^d$ given $n$ data points, each obtained by perturbing a latent point
in $K$}. This problem does not seem to have been considered in the literature.
The most important contribution of this paper is to show that the latent
$k-$polytope problem admits an efficient algorithm under deterministic
assumptions which naturally hold in Latent variable models considered in this
paper. ur algorithm runs in time $O^*(k\; \mbox{nnz})$ matching the best
running time of algorithms in special cases considered here and is better when
the data is sparse, as is the case in applications. An important novelty of the
algorithm is the introduction of \emph{subset smoothed polytope}, $K'$, the
convex hull of the ${n\choose \delta n}$ points obtained by averaging all
$\delta n$ subsets of the data points, for a given $\delta \in (0,1)$. We show
that $K$ and $K'$ are close in Hausdroff distance. Among the consequences of
our algorithm are the following: (a) MMSB Models and Topic Models: the first
quasi-input-sparsity time algorithm for parameter estimation for $k \in
O^*(1)$, (b) Adversarial Clustering: In $k-$means, if, an adversary is allowed
to move many data points from each cluster an arbitrary amount towards the
convex hull of the centers of other clusters, our algorithm still estimates
cluster centers well.
