We present an approach to backpropagating through minimal problem solvers in
end-to-end neural network training. Traditional methods relying on manually
constructed formulas, finite differences, and autograd are laborious,
approximate, and unstable for complex minimal problem solvers. We show that
using the Implicit function theorem to calculate derivatives to backpropagate
through the solution of a minimal problem solver is simple, fast, and stable.
We compare our approach to (i) using the standard autograd on minimal problem
solvers and relate it to existing backpropagation formulas through SVD-based
and Eig-based solvers and (ii) implementing the backprop with an existing
PyTorch Deep Declarative Networks (DDN) framework. We demonstrate our technique
on a toy example of training outlier-rejection weights for 3D point
registration and on a real application of training an outlier-rejection and
RANSAC sampling network in image matching. Our method provides $100\%$
stability and is 10 times faster compared to autograd, which is unstable and
slow, and compared to DDN, which is stable but also slow.

我们提出了一种通过最小问题求解器进行反向传播的方法，使用隐函数定理计算倒数，以实现对最小问题求解器的快速、稳定的训练。