Near-optimal sample complexity bounds for learning Latent $k-$polytopes and applications to Ad-Mixtures

Deriving Optimal bounds on Sample Complexity of Latent Variable models is an active area of research. Recently such bounds were obtained for Mixture of Gaussians \cite{HSNCAY18}, no such results are known for Ad-mixtures, a generalization of Mixture distributions. In this paper we show that $O^*(dk/m)$ samples are sufficient to learn each of $k-$ topic vectors of LDA, a popular Ad-mixture model, with vocabulary size $d$ and $m\in \Omega(1)$ words per document, to any constant error in $L_1$ norm. The result is a corollary of the major contribution of this paper: the first sample complexity upper bound for the problem (introduced in \cite{BK20}) of learning the vertices of a Latent $k-$ Polytope in $\RR^d$, given perturbed points from it. The bound, $O^*(dk/\beta)$, is optimal and linear in number of parameters. It applies to many stochastic models including a broad class Ad-mixtures. To demonstrate the generality of the approach we specialize the setting to Mixed Membership Stochastic Block Models(MMSB) and show for the first time that if an MMSB has $k$ blocks, the sample complexity is $O^*(k^2)$ under usual assumptions.