On the Robustness of the Successive Projection Algorithm
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909887215173632 |
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| author | Barbarino, Giovanni Gillis, Nicolas |
| author_facet | Barbarino, Giovanni Gillis, Nicolas |
| contents | The successive projection algorithm (SPA) is a workhorse algorithm to learn the $r$ vertices of the convex hull of a set of $(r-1)$-dimensional data points, a.k.a. a latent simplex, which has numerous applications in data science. In this paper, we revisit the robustness to noise of SPA and several of its variants. In particular, when $r \geq 3$, we prove the tightness of the existing error bounds for SPA and for two more robust preconditioned variants of SPA. We also provide significantly improved error bounds for SPA, by a factor proportional to the conditioning of the $r$ vertices, in two special cases: for the first extracted vertex, and when $r \leq 2$. We then provide further improvements for the error bounds of a translated version of SPA proposed by Arora et al. (''A practical algorithm for topic modeling with provable guarantees'', ICML, 2013) in two special cases: for the first two extracted vertices, and when $r \leq 3$. Finally, we propose a new more robust variant of SPA that first shifts and lifts the data points in order to minimize the conditioning of the problem. We illustrate our results on synthetic data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16195 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Robustness of the Successive Projection Algorithm Barbarino, Giovanni Gillis, Nicolas Numerical Analysis Data Structures and Algorithms Machine Learning The successive projection algorithm (SPA) is a workhorse algorithm to learn the $r$ vertices of the convex hull of a set of $(r-1)$-dimensional data points, a.k.a. a latent simplex, which has numerous applications in data science. In this paper, we revisit the robustness to noise of SPA and several of its variants. In particular, when $r \geq 3$, we prove the tightness of the existing error bounds for SPA and for two more robust preconditioned variants of SPA. We also provide significantly improved error bounds for SPA, by a factor proportional to the conditioning of the $r$ vertices, in two special cases: for the first extracted vertex, and when $r \leq 2$. We then provide further improvements for the error bounds of a translated version of SPA proposed by Arora et al. (''A practical algorithm for topic modeling with provable guarantees'', ICML, 2013) in two special cases: for the first two extracted vertices, and when $r \leq 3$. Finally, we propose a new more robust variant of SPA that first shifts and lifts the data points in order to minimize the conditioning of the problem. We illustrate our results on synthetic data. |
| title | On the Robustness of the Successive Projection Algorithm |
| topic | Numerical Analysis Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2411.16195 |