Improved Algorithm and Bounds for Successive Projection
<|reference_start|>Improved Algorithm and Bounds for Successive Projection: 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.<|reference_end|>
@article{jin2024improved,
    title={Improved Algorithm and Bounds for Successive Projection},
    author={Jiashun Jin, Zheng Tracy Ke, Gabriel Moryoussef, Jiajun Tang, Jingming
  Wang},
    journal={arXiv preprint arXiv:2403.11013},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.11013},
    primaryClass={cs.LG math.ST stat.TH}
}