Improved Algorithm and Bounds for Successive Projection
Jiashun Jin, Zheng Tracy Ke, Gabriel Moryoussef, Jiajun Tang, Jingming, Wang
TL;DR
This paper introduces pp-SPA, an enhanced vertex hunting algorithm that combines projection and denoising steps, providing faster and more accurate results than the traditional SPA, especially under high noise conditions.
Contribution
The paper proposes pp-SPA, a novel variant of SPA that improves vertex estimation accuracy and speed by incorporating denoising, along with new error bounds for SPA.
Findings
pp-SPA outperforms SPA in noisy environments
Faster convergence rates for pp-SPA
Improved non-asymptotic bounds for SPA
Abstract
Given a -vertex simplex in a -dimensional space, suppose we measure points on the simplex with noise (hence, some of the observed points fall outside the simplex). Vertex hunting is the problem of estimating the 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.
Peer Reviews
Decision·ICLR 2024 poster
I like the pp-SPA algorithm a lot, and it feels like a very natural idea. It's an added bonus that the authors were able to get tighter theoretical bounds, which while admittedly are too complicated sometimes to compare fairly to classical SPA, are clearly tighter. It's not clear to me at all when the bounds for pp-SPA beat the bounds for classical SPA, simply because of how complicated the bounds are. I was able to follow most proofs as a non-expert, so the **proofs** are well written (the ac
1) For some reason, the authors do not compare pp-SPA with any robust SPA. I found many results in the literature on robust SPA, so it seems bizare it is not included in the experimental section here. Please try to include at least one robust variant of SPA/similar algorithm in the experimental section. 2) The writing can be dramatically improved. See below minor points, but if this is accepted, please spent a few iterations on improving the writing. It's extremely terse at time. 3) You repr
The paper gives ample motivating examples for which the problem is relevant. The paper makes attempts to motivate the given bounds. The algorithm itself seems intuitive.
The bounds given in the paper, e.g. Theorem 2, are very complex; they contain multiple terms and are subject to many conditions. It is hard to understand whether those conditions are restrictive, or whether the bounds are significant. I think the presentation of the problem could be expanded upon.
I believe, the paper brings potentially several strong contributions: 1) The authors propose algorithm seems to practically make a lot of sense 2) The theoretical results are novel and non-trivial. The authors exploit the specific Gaussianity of the noise to be able to derive these bounds 3) The experiments are limited, but they support the statements of the paper
There are some considerable difficulties I have with the paper: 1) The writing and the structure of the text should be improved. For example: * It is not clear to me what the illustration in Figure 1 shows? Is it obvious that one of these is better and what does ``idea simplex'' denote? * At times a statement is given without citation, e.g. you say in "Our contributions.", pg 2 that: "since the SPA is greedy algorithm, it tends to be biased outward bound.", but it is not clear where this can be
Code & Models
Videos
Taxonomy
TopicsRobotic Mechanisms and Dynamics · Manufacturing Process and Optimization · Advanced Numerical Analysis Techniques