Conditional Gradients for the Approximate Vanishing Ideal
This work addresses the challenge of handling noise in data for polynomial structure extraction, offering a theoretically motivated method for machine learning applications, though it appears incremental as it builds on existing generator-constructing methods.
The paper tackles the problem of constructing generators for the approximate vanishing ideal from noisy data, introducing the PCGAVI algorithm that produces sparse generators, which can be used as a feature map for tasks like supervised learning with linear classifiers.
The vanishing ideal of a set of points $X\subseteq \mathbb{R}^n$ is the set of polynomials that evaluate to $0$ over all points $\mathbf{x} \in X$ 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.