Improved Algorithm and Bounds for Successive Projection
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914717565452288 |
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| author | Jin, Jiashun Ke, Zheng Tracy Moryoussef, Gabriel Tang, Jiajun Wang, Jingming |
| author_facet | Jin, Jiashun Ke, Zheng Tracy Moryoussef, Gabriel Tang, Jiajun Wang, Jingming |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_11013 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved Algorithm and Bounds for Successive Projection Jin, Jiashun Ke, Zheng Tracy Moryoussef, Gabriel Tang, Jiajun Wang, Jingming Machine Learning Statistics Theory 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. |
| title | Improved Algorithm and Bounds for Successive Projection |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2403.11013 |