Improved algorithm and bounds for successive projection

Consider a $K$-vertex simplex in a $d$-dimensional space. We measure $n$ points on the simplex, but due to the measurement noise, some of the observed points fall outside the simplex. The interest is vertex hunting (i.e., estimating the vertices of the simplex). The successive projection algorithm (SPA) is one of the most popular approaches to vertex hunting, but it is vulnerable to noise and outliers, and may perform unsatisfactorily. We propose pseudo-point SPA (pp-SPA) as a new approach to vertex hunting. The approach contains two novel ideas (a projection step and a denoise step) and generates roughly $n$ pseudo-points, which can be fed in to SPA for vertex hunting. For theory, we first derive an improved non-asymptotic bound for the orthodox SPA, and then use the result to derive the bounds for pp-SPA. Compared with the orthodox SPA, pp-SPA has a faster rate and more satisfactory numerical performance in a broad setting. The analysis is quite delicate: the non-asymptotic bound is hard to derive, and we need precise results on the extreme values of (possibly) high-dimensional random vectors.