Papers › Robust Vertex Enumeration for Convex Hulls in High Dimensions
Robust Vertex Enumeration for Convex Hulls in High Dimensions
Pranjal Awasthi, Bahman Kalantari, Yikai Zhang
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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 S of all K vertices of conv(S), the convex hull of S. If Γ_* is the minimum of the distances from each vertex to the convex hull of the remaining vertices, given any γ≤γ_* = Γ_*/R, R the diameter of S, AVTA computes S in O(nK(m+ γ⁻²)) operations. If γ_* is unknown but K is known, AVTA computes S in O(nK(m+ γ_*⁻²)) log(γ_*⁻¹) operations. More generally, given t ∈(0,1), AVTA computes a subset Sᵗ of S in O(n |Sᵗ|(m+ t⁻²)) operations, where the distance between any p ∈conv(S) to conv(Sᵗ) is at most t R. Next we consider AVTA where input is S_ε, an ε perturbation of S. Assuming a bound on ε 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 S_ε. We also analyze AVTA under random projections of S or S_ε. 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~\cite{arora2013practical, bansal2014provable}. For non-negative matrix AVTA is competitive with existing methods~\cite{arora2012computing}. Empirically AVTA is robust and can handle larger amounts of noise than existing methods.
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