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# Finding k in Latent $k-$ polytope

*Proceedings of the 38th International Conference on Machine Learning*, PMLR 139:894-903, 2021.

#### Abstract

The recently introduced Latent $k-$ Polytope($\LkP$) encompasses several stochastic Mixed Membership models including Topic Models. The problem of finding $k$, the number of extreme points of $\LkP$, is a fundamental challenge and includes several important open problems such as determination of number of components in Ad-mixtures. This paper addresses this challenge by introducing Interpolative Convex Rank(\INR) of a matrix defined as the minimum number of its columns whose convex hull is within Hausdorff distance $\varepsilon$ of the convex hull of all columns. The first important contribution of this paper is to show that under \emph{standard assumptions} $k$ equals the \INR of a \emph{subset smoothed data matrix} defined from Data generated from an $\LkP$. The second important contribution of the paper is a polynomial time algorithm for finding $k$ under standard assumptions. An immediate corollary is the first polynomial time algorithm for finding the \emph{inner dimension} in Non-negative matrix factorisation(NMF) with assumptions which are qualitatively different than existing ones such as \emph{Separability}. %An immediate corollary is the first polynomial time algorithm for finding the \emph{inner dimension} in Non-negative matrix factorisation(NMF) with assumptions considerably weaker than \emph{Separability}.