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.

翻译：给定 $d$ 维空间中的一个 $K$ 顶点单纯形，假设我们在带噪声的情况下测量该单纯形上的 $n$ 个点（因此，部分观测点落在单纯形外）。顶点狩猎问题旨在估计该单纯形的 $K$ 个顶点。一种流行的顶点狩猎算法是逐次投影算法（SPA）。然而，在强噪声或存在异常值的情况下，SPA 的表现不尽如人意。我们提出了伪点逐次投影算法（pp-SPA）。该算法通过投影步骤与去噪步骤生成伪点，并将其输入 SPA 以进行顶点狩猎。我们借助（可能为）高维随机向量的极值理论，推导了 pp-SPA 的误差界。结果表明，pp-SPA 比 SPA 具有更快的收敛速度和更好的数值表现。我们的分析还包含了原始 SPA 的一个改进的非渐近界，该结果具有独立的研究价值。