The paradigm of differentiable programming has significantly enhanced the scope of machine learning via the judicious use of gradient-based optimization. However, standard differentiable programming methods (such as autodiff) typically require that the machine learning models be differentiable, limiting their applicability. Our goal in this paper is to use a new, principled approach to extend gradient-based optimization to functions well modeled by splines, which encompass a large family of piecewise polynomial models. We derive the form of the (weak) Jacobian of such functions and show that it exhibits a block-sparse structure that can be computed implicitly and efficiently. Overall, we show that leveraging this redesigned Jacobian in the form of a differentiable "layer" in predictive models leads to improved performance in diverse applications such as image segmentation, 3D point cloud reconstruction, and finite element analysis.

翻译：不同的编程模式通过明智地使用基于梯度的优化,极大地扩大了机器学习的范围。然而,标准的可区分的编程方法(如自动调试)通常要求机器学习模式具有差异性,限制其适用性。我们本文件的目标是采用新的原则性办法,将基于梯度的优化扩展至由样条模型精心建模的功能,这包括一大批小块式多元模型。我们从这种功能的(微弱的)雅各克语形式中推导出一个可以隐含和有效计算的结构。总体而言,我们表明,以可不同“层”的预测模型形式利用这个重新设计的雅各克人,可以改善图解、三维点云重组和有限要素分析等多种应用的性能。