Improved Algorithm and Bounds for Successive Projection

TL;DR

This work proposes pseudo-point SPA, which uses a projection step and a denoise step to generate pseudo-points and feed them into SPA for vertex hunting, and derives error bounds for pp-SPA, leveraging on extreme value theory of (possibly) high-dimensional random vectors.

Abstract

Given a $K$-vertex simplex in a $d$-dimensional space, suppose we measure $n$ points on the simplex with noise (hence, some of the observed points fall outside the simplex). Vertex hunting is the problem of estimating the $K$ vertices of the simplex. A popular vertex hunting algorithm is successive projection algorithm (SPA). However, SPA is observed to perform unsatisfactorily under strong noise or outliers. We propose pseudo-point SPA (pp-SPA). It uses a projection step and a denoise step to generate pseudo-points and feed them into SPA for vertex hunting. We derive error bounds for pp-SPA, leveraging on extreme value theory of (possibly) high-dimensional random vectors. The results suggest that pp-SPA has faster rates and better numerical performances than SPA. Our analysis includes an improved non-asymptotic bound for the original SPA, which is of independent interest.

Figures (5)

• Figure 1: A numerical example ($d$=$2$, $K$=$3$).
• Figure 2: A toy example to show the difference between $\beta(X)$ and $\beta_{\text{new}}(X,V)$, where $\beta(X)=\max_i \|\epsilon_i\|$, and $\beta_{\text{new}}(X, V)\leq \max_{i\notin\{2,5\}}\|\epsilon_i\|$.
• Figure 3: Performances of SPA, P-SPA, D-SPA, and pp-SPA in Experiment 1-3.
• Figure 4: An illustration of the simplicial neighborhoods and ${\cal V}(\epsilon_0, h_0)$.
• Figure 5: Factor of improvement of our bound over orthodox spa as the true simplex moves away from origin by a distance $a$.

Theorems & Definitions (23)

• Lemma 1
• Lemma 2: gillis2013fast, orthodox SPA
• Lemma 3
• Theorem 1
• Lemma 4
• Lemma 5
• Theorem 2
• Theorem 3
• Theorem B.1
• Lemma B.1