Differentiable Spline Approximations
Differentiable Spline Approximations
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发表时间：
2021-10
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通讯作者：
Minsu Cho;Aditya Balu;Ameya Joshi;Anjana Prasad;Biswajit Khara;S. Sarkar;B. Ganapathysubramanian;A. Krishnamurthy;C. Hegde
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作者：
Minsu Cho;Aditya Balu;Ameya Joshi;Anjana Prasad;Biswajit Khara;S. Sarkar;B. Ganapathysubramanian;A. Krishnamurthy;C. Hegde
可微分编程范式通过明智地使用基于梯度的优化，显着扩大了机器学习的范围。然而，标准的可微分编程方法（例如 autodiff）通常要求机器学习模型是可微分的，从而限制了它们的适用性。我们在本文中的目标是使用一种新的、有原则的方法将基于梯度的优化扩展到通过样条曲线很好地建模的函数，其中包含大量分段多项式模型。我们推导了此类函数的（弱）雅可比行列式的形式，并表明它呈现出可以隐式有效计算的块稀疏结构。总体而言，我们表明，在预测模型中以可微“层”的形式利用这种重新设计的雅可比行列式可以提高图像分割、3D 点云重建和有限元分析等各种应用中的性能。
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.