Abstract: Given data vectors X₁,…X_n ∈ ℝ^r, where X_i is a noisy observation of $X_i^{*}$, and $X_1^{*}$,…$X_n^{*}$ are contained in an unknown simplex with K vertices, vertex hunting (VH) is the problem of estimating the vertices of the true simplex. VH is a building block of several algorithms in hyperspectral remote sensing, soft clustering, topic modeling, and network mixed membership estimation. The popular VH algorithms are susceptible to outliers, whose estimation errors are governed by ${max}_i\parallel X_i-X_i^{*}\parallel$. We propose a robust VH algorithm that properly shrinks estimated vertices towards the interior of data cloud, so as to mitigate the effect of outliers. The level of shrinkage is determined by maximizing a pseudo likelihood and has no tuning parameter. We show that, when the barycentric coordinates of $X_1^{*}$,…,$X_n^{*}$ come from a Dirichlet distribution, the proposed method has a faster rate of convergence than several popular VH algorithms.

Key words and phrases: Admixture, archetypal analysis, Dirichlet distribution, endmember extraction, gradient descent, linear unmixing, mixed membership estimation, nonnegative matrix factorization, pseudo likelihood, topic modeling.