Supplemental Material for “ A Practical Algorithm for Topic Modeling with Provable Guarantees ”

(a) (b) (c) (d) Figure 1. Illustration of the Algorithm Recall that the correctness of the algorithm depends on the following Lemma: Lemma 1.1. The point d j found by the algorithm must be δ = O(/γ 2) close to some vertex v i. In particular , the corresponding a j O(/γ 2)-covers v i. In order to prove this Lemma, we first show that even if previously found vertices are only δ close to some vertices, there is still another vertex that is far from the span of previously found vertices. Lemma 1.2. Suppose all previously found vertices are O(/γ 2) close to distinct vertices, there is a vertex v i whose distance from span(S) is at least γ/2. In order to prove Lemma 1.2, we use a volume argument. First we show that the volume of a robust simplex cannot change by too much when the vertices are perturbed. } are the vertices of a γ-robust simplex S. Let S be a simplex with ver-tices {v √ Kδ < γ the volume of the two simplices satisfy vol(S)(1 − 2δ/γ) K−1 ≤ vol(S) ≤ vol(S)(1 + 4δ/γ) K−1. Proof: As the volume of a simplex is proportional to the determinant of a matrix whose columns are the edges of the simplex, we first show the following perturbation bound for determinant.