# Dual-to-kernel learning with ideals

arXiv: https://arxiv.org/abs/1402.0099 · HTML: https://mlsanity.com/paper/1402.0099/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Feb 2014.

## Built on by (most cited first, 5 of 5)

- [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)
- [Vanishing Component Analysis with Contrastive Normalization](https://mlsanity.com/paper/2210.16171/) (2022)

## Abstract (arXiv)

In this paper, we propose a theory which unifies kernel learning and symbolic algebraic methods. We show that both worlds are inherently dual to each other, and we use this duality to combine the structure-awareness of algebraic methods with the efficiency and generality of kernels. The main idea lies in relating polynomial rings to feature space, and ideals to manifolds, then exploiting this generative-discriminative duality on kernel matrices. We illustrate this by proposing two algorithms, IPCA and AVICA, for simultaneous manifold and feature learning, and test their accuracy on synthetic and real world data.