Online Stability Improvement of Groebner Basis Solvers using Deep Learning

TL;DR

The paper addresses instability in Groebner-basis solvers caused by fixed elimination templates by revealing that, for permutation-invariant polynomial systems, variable reordering corresponds to column permutations and can be exploited to reuse templates. It introduces a small neural classifier that predicts the best permutation among $n!$ possibilities from the original coefficients, enabling online, low-overhead improvement of numerical stability for generic dense solvers and a UPnP camera resectioning solver. The approach relies on a formal definition of permutation-invariant polynomials, synthetic data-driven training, and permutation-aware augmentation to maintain invariance. The results demonstrate substantial stability and accuracy gains in diverse dense polynomial problems and in a practical geometric vision application, highlighting the method's potential to enhance Groebner-based solvers in vision tasks and beyond.

Abstract

Over the past decade, the Gröbner basis theory and automatic solver generation have lead to a large number of solutions to geometric vision problems. In practically all cases, the derived solvers apply a fixed elimination template to calculate the Gröbner basis and thereby identify the zero-dimensional variety of the original polynomial constraints. However, it is clear that different variable or monomial orderings lead to different elimination templates, and we show that they may present a large variability in accuracy for a certain instance of a problem. The present paper has two contributions. We first show that for a common class of problems in geometric vision, variable reordering simply translates into a permutation of the columns of the initial coefficient matrix, and that -- as a result -- one and the same elimination template can be reused in different ways, each one leading to potentially different accuracy. We then prove that the original set of coefficients may contain sufficient information to train a classifier for online selection of a good solver, most notably at the cost of only a small computational overhead. We demonstrate wide applicability at the hand of generic dense polynomial problem solvers, as well as a concrete solver from geometric vision.

Figures (8)

• Figure 1: Error distribution over many random instances of the UPnP problem. Each curve represents the distribution obtained by one of the variable orderings. The red curve furthermore indicates the obtainable result if---for each instance---the best variable ordering is chosen.
• Figure 2: Permutation of the coefficient matrix $\mathbf{C}$ and back-permutation of the solutions $\mathcal{S}=\{\mathbf{x}_1^{*},\ldots,\mathbf{x}_s^{*}\}$ for a potential improvement of the numerical stability of an elimination template.
• Figure 3: Permutation-aware Gröbner basis solver with added classifier able to predict a good permutation $\mathbf{P}^{*}$ directly from the original coefficient matrix $\mathbf{C}$.
• Figure 4: Architecture of our neural network for predicting and selecting a good permutation.
• Figure 5: Training data augmentation for permutation-invariant classification performance.