Appendix A Proofs and derivations Theorem 1. The Jacobian of the operation F with respect to x 2 R

The proof follows similar arguments as in Proposition 4 from Blondel et al. [ 2020 ]. We now derive differentiable forms of generalized piecewise d -polynomial regression, which is used in applications such as spline fittings. Our 1D piecewise spline approximation can be (heuristically) extended to 2D data. We consider the problem of image segmentation, which can be viewed as representing the domain of an image into a disjoint union of subsets. Instead, we leverage connected-component algorithms (such as Hoshen-Kopelman, or other, techniques [ Wu et al., 2005 ]) to produce a partition, and the predicted output is a piecewise constant image with C.2 NURBS derivatives We rewrite the NURBS formulation as follows: S ( u, v)= NR ( u, v) w ( u, v) (20) where, NR ( u, v)= For simplicity, we will stick to 1D NURBS curves.