Differentiable Spline Approximations

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Cho, Minsu
Balu, Aditya
Joshi, Ameya
Prasad, Anjana Deva
Khara, Biswajit
Sarkar, Soumik
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Krishnamurthy, Adarsh
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Mechanical Engineering
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Electrical and Computer Engineering

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  • Department of Electrical Engineering and Computer Engineering (1985-1995)

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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.
This is a proceeding preprint from Cho, Minsu, Aditya Balu, Ameya Joshi, Anjana Deva Prasad, Biswajit Khara, Soumik Sarkar, Baskar Ganapathysubramanian, Adarsh Krishnamurthy, and Chinmay Hegde. "DIFFERENTIABLE SPLINE APPROXIMATIONS." arXiv preprint arXiv:2110.01532 (2021). doi: https://doi.org/10.48550/arXiv.2110.01532. Copyright 2021 The Authors.