On the Robustness of the Successive Projection Algorithm

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Abstract
The successive projection algorithm (SPA) is a workhorse algorithm to learn the [Formula: see text] vertices of the convex hull of a set of [Formula: see text]-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 [Formula: see text], 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 [Formula: see text] vertices, in two special cases: for the first extracted vertex and when [Formula: see text]. We then provide further improvements for the error bounds of a translated version of SPA proposed by Arora et al. [ Proceedings of the International Conference on Machine Learning, 2013, pp. 280–288] in two special cases: for the first two extracted vertices and when [Formula: see text]. 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. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SIMAX and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://gitlab.com/ngillis/robustspa . [Formula: see text]
Funding Information
  • Istituto Nazionale di Alta Matematica
  • H2020 European Research Council (101085607)

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