Online Stability Improvement of Groebner Basis Solvers using Deep Learning

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.