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Conditional Gradients for the Approximate Vanishing Ideal

Elias Wirth, Sebastian Pokutta

TL;DR

The paper addresses constructing a finite, sparse set of generators for the (approximate) vanishing ideal ${\\mathcal{I}}_X$ of a data set ${X} \,\subseteq {\\mathbb{R}}^n$ to enable robust feature maps for linear classifiers. It introduces PCGAVI, which solves constrained convex subproblems via Pairwise Conditional Gradients (PCG) to produce ${\\tau}$-bounded, sparse generators, together with a rigorous generalization framework. The Oracle Approximate Vanishing Ideal Algorithm (OAVI) underpins PCGAVI by iteratively building an order-ideal border and certifying candidates with an ORACLE, yielding theoretical guarantees and explicit complexity bounds. Empirically, PCGAVI achieves competitive classification performance with notably sparse transformations and faster test-time behavior on large datasets, while border choices influence generator counts and sparsity; these results position PCGAVI as a theoretically grounded, scalable alternative to prior methods such as ABM, AVI, and VCA.

Abstract

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.

Conditional Gradients for the Approximate Vanishing Ideal

TL;DR

The paper addresses constructing a finite, sparse set of generators for the (approximate) vanishing ideal of a data set to enable robust feature maps for linear classifiers. It introduces PCGAVI, which solves constrained convex subproblems via Pairwise Conditional Gradients (PCG) to produce -bounded, sparse generators, together with a rigorous generalization framework. The Oracle Approximate Vanishing Ideal Algorithm (OAVI) underpins PCGAVI by iteratively building an order-ideal border and certifying candidates with an ORACLE, yielding theoretical guarantees and explicit complexity bounds. Empirically, PCGAVI achieves competitive classification performance with notably sparse transformations and faster test-time behavior on large datasets, while border choices influence generator counts and sparsity; these results position PCGAVI as a theoretically grounded, scalable alternative to prior methods such as ABM, AVI, and VCA.

Abstract

The vanishing ideal of a set of points is the set of polynomials that evaluate to over all points 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.
Paper Structure (40 sections, 15 theorems, 16 equations, 1 figure, 3 tables, 3 algorithms)

This paper contains 40 sections, 15 theorems, 16 equations, 1 figure, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Works
  4. Outline
  5. Preliminaries
  6. Convex Optimization
  7. The Oracle Approximate Vanishing Ideal Algorithm (OAVI)
  8. Algorithm Overview
  9. Input
  10. Initialization
  11. Line \ref{['alg:while_loop']}
  12. While-Loop
  13. Termination
  14. Analysis
  15. Generator Construction
  16. ...and 25 more sections

Key Result

Theorem 3.1

Let $X = \{{\mathbf{x}}_1, \ldots, {\mathbf{x}}_m\} \subseteq \mathbb{R}^n$, $\psi\geq 0$, ${\mathcal{O}}\xspace = \{t_1, \ldots t_k\}_\sigma \subseteq {\mathcal{T}}\xspace$, $t\in {\mathcal{T}}\xspace$ such that $t >_\sigma t_i$ for all $i\in \{1, \ldots, k\}$, and ${\mathbf{d}}$ as in eq:cop. Ther

Figures (1)

  • Figure 1: Comparison of the number of generators constructed with PCGAVI and ABM for the border-\ref{['eq:gb']} and the border-\ref{['eq:bb']}, averaged over ten random runs with shaded standard deviations. Running algorithms with with the border-\ref{['eq:bb']} often leads to the construction of more generators than with the border-\ref{['eq:gb']}. See Section \ref{['sec:border_types_G']} for details on the setup.

Theorems & Definitions (33)

  • Definition 2.1: Approximately vanishing polynomial
  • Definition 2.2: Leading term (coefficient)
  • Definition 2.3: Approximate vanishing ideal
  • Theorem 3.1: Certificate
  • Lemma 4.1: kreuzer2000computational
  • Definition 4.2: Border
  • Proposition 4.3
  • proof
  • Theorem 4.4: Maximality
  • proof
  • ...and 23 more