Distributions for Compositionally Differentiating Parametric Discontinuities
Computations in computer graphics, robotics, and probabilistic inference often require differentiating integrals with discontinuous integrands. Popular differentiable programming languages do not support the differentiation of these integrals. We introduce a differentiable programming language, Potto, that is the first to differentiate parametric discontinuities under integration, while supporting first-order functions and compositional evaluation. We extend distribution theory to provide semantic definitions for the derivatives of programs of interest. Using this theory, we present a denotational semantics and a distributional derivative interpretation and show the two accord. We provide operational semantics for both the program and the distributional derivative and soundness theorems.