# Gradient Boosts the Approximate Vanishing Ideal

arXiv: https://arxiv.org/abs/1911.04174 · HTML: https://mlsanity.com/paper/1911.04174/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Nov 2019.

## Built on by (most cited first, 8 of 8)

- [Learning to Compute Gr"obner Bases](https://mlsanity.com/paper/2311.12904/) (2023)
- [Computational Algebra with Attention: Transformer Oracles for Border Basis Algorithms](https://mlsanity.com/paper/2505.23696/) (2025)
- [Approximate Vanishing Ideal Computations at Scale](https://mlsanity.com/paper/2207.01236/) (2022)
- [Border basis computation with gradient-weighted normalization](https://mlsanity.com/paper/2101.00401/) (2021)
- [Evolving Architectures with Gradient Misalignment toward Low Adversarial Transferability](https://mlsanity.com/paper/2109.05919/) (2021)
- [Conditional Gradients for the Approximate Vanishing Ideal](https://mlsanity.com/paper/2202.03349/) (2022)
- [Vanishing Component Analysis with Contrastive Normalization](https://mlsanity.com/paper/2210.16171/) (2022)
- [Diagnosing Faults in Reinforcement Learning Simulators and World Models with Canonical Polynomial Invariants](https://mlsanity.com/paper/2609.13194/) (2026)

## Abstract (arXiv)

In the last decade, the approximate vanishing ideal and its basis construction algorithms have been extensively studied in computer algebra and machine learning as a general model to reconstruct the algebraic variety on which noisy data approximately lie. In particular, the basis construction algorithms developed in machine learning are widely used in applications across many fields because of their monomial-order-free property; however, they lose many of the theoretical properties of computer-algebraic algorithms. In this paper, we propose general methods that equip monomial-order-free algorithms with several advantageous theoretical properties. Specifically, we exploit the gradient to (i) sidestep the spurious vanishing problem in polynomial time to remove symbolically trivial redundant bases, (ii) achieve consistent output with respect to the translation and scaling of input, and (iii) remove nontrivially redundant bases. The proposed methods work in a fully numerical manner, whereas existing algorithms require the awkward monomial order or exponentially costly (and mostly symbolic) computation to realize properties (i) and (iii). To our knowledge, property (ii) has not been achieved by any existing basis construction algorithm of the approximate vanishing ideal.