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In this paper we show that a large class of Latent variable models, such asMixed Membership Stochastic Block(MMSB) Models, Topic Models, and AdversarialClustering, can be unified through a geometric perspective, replacing modelspecific assumptions and algorithms for individual models. The geometricperspective 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 pointin $K$}. 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 deterministicassumptions which naturally hold in Latent variable models considered in thispaper. ur algorithm runs in time $O^*(k\; \mbox{nnz})$ matching the bestrunning time of algorithms in special cases considered here and is better whenthe data is sparse, as is the case in applications. An important novelty of thealgorithm is the introduction of \emph{subset smoothed polytope}, $K'$, theconvex 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 showthat $K$ and $K'$ are close in Hausdroff distance. Among the consequences ofour algorithm are the following: (a) MMSB Models and Topic Models: the firstquasi-input-sparsity time algorithm for parameter estimation for $k \inO^*(1)$, (b) Adversarial Clustering: In $k-$means, if, an adversary is allowedto move many data points from each cluster an arbitrary amount towards theconvex hull of the centers of other clusters, our algorithm still estimatescluster centers well.

Finding a latent k-simplex in O(k . nnz(data)) time via Subset Smoothing