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.

Understanding sequential information is a fundamental task for artificial intelligence. Current neural networks attempt to learn spatial and temporal information as a whole, limited their abilities to represent large scale spatial representations over long-range sequences. Here, we introduce a new modeling strategy called Semi-Coupled Structure (SCS), which consists of deep neural networks that decouple the complex spatial and temporal concepts learning. Semi-Coupled Structure can learn to implicitly separate input information into independent parts and process these parts respectively. Experiments demonstrate that a Semi-Coupled Structure can successfully annotate the outline of an object in images sequentially and perform video action recognition. For sequence-to-sequence problems, a Semi-Coupled Structure can predict future meteorological radar echo images based on observed images. Taken together, our results demonstrate that a Semi-Coupled Structure has the capacity to improve the performance of LSTM-like models on large scale sequential tasks.