Robust Vertex Enumeration for Convex Hulls in High Dimensions

Computation of the vertices of the convex hull of a set S of n points in ℝ m is a fundamental problem in computational geometry, optimization, machine learning and more. We present "All Vertex Triangle Algorithm" (AVTA), a robust and efficient algorithm for computing the subset overline S of all K vertices of conv(S), the convex hull of S. If Gamma_* is the minimum of the distances from each vertex to the convex hull of the remaining vertices, given any γ ≤ γ_* = Gamma_*/R, R the diameter of S, AVTA computes overline S in O(nK(m+ γ-2)) operations. If γ_* is unknown but K is known, AVTA computes overline S in O(nK(m+ γ_*-2)) log(γ_*-1) operations. More generally, given t ∈ (0,1), AVTA computes a subset overline St of overline S in O(n |overline St|(m+ t-2)) operations, where the distance between any p ∈ conv(S) to conv(overline St) is at most t R. Next we consider AVTA where input is S_varepsilon, an varepsilon perturbation of S. Assuming a bound on varepsilon in terms of the minimum of the distances of vertices of conv(S) to the convex hull of the remaining point of S, we derive analogous complexity bounds for computing overline S_varepsilon. We also analyze AVTA under random projections of S or S_varepsilon. Finally, via AVTA we design new practical algorithms for two popular machine learning problems: topic modeling and non-negative matrix factorization. For topic models AVTA leads to significantly better reconstruction of the topic-word matrix than state of the art approaches~citearora2013practical, bansal2014provable. For non-negative matrix AVTA is competitive with existing methods~citearora2012computing. Empirically AVTA is robust and can handle larger amounts of noise than existing methods.