Online Stability Improvement of Groebner Basis Solvers using Deep
  Learning
Wanting Xu Lan Hu Manolis C. Tsakiris Laurent Kneip
Over the past decade, the Gr\"obner 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 Gr\"obner 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.