2022/02/07 by Elias Wirth, Sebastian Pokutta, Wirth, Elias +1
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Commutative Algebra and Its Applications #FOS: Computer and information sciences #Machine Learning (cs.LG) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2202.03349
openalex publication_date 2022/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The vanishing ideal of a set of points X⊆ ℝn is the set of polynomials that evaluate to 0 over all points x ∈ X and admits an efficient representation by a finite set of polynomials called generators. To accommodate the noise in the data set, we introduce the pairwise conditional gradients approximate vanishing ideal algorithm (PCGAVI) that constructs a set of generators of the approximate vanishing ideal. The constructed generators capture polynomial structures in data and give rise to a feature map that can, for example, be used in combination with a linear classifier for supervised learning. In PCGAVI, we construct the set of generators by solving constrained convex optimization problems with the pairwise conditional gradients algorithm. Thus, PCGAVI not only constructs few but also sparse generators, making the corresponding feature transformation robust and compact. Furthermore, we derive several learning guarantees for PCGAVI that make the algorithm theoretically better motivated than related generator-constructing methods.