# Distribution Theoretic Semantics for Non-Smooth Differentiable Programming

arXiv: https://arxiv.org/abs/2207.05946 · HTML: https://mlsanity.com/paper/2207.05946/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jul 2022.

## Built on by (most cited first, 1 of 1)

- [Smoothing Methods for Automatic Differentiation Across Conditional Branches](https://mlsanity.com/paper/2310.03585/) (2023)

## Abstract (arXiv)

With the wide spread of deep learning and gradient descent inspired optimization algorithms, differentiable programming has gained traction. Nowadays it has found applications in many different areas as well, such as scientific computing, robotics, computer graphics and others. One of its notoriously difficult problems consists in interpreting programs that are not differentiable everywhere. In this work we define λδ, a core calculus for non-smooth differentiable programs and define its semantics using concepts from distribution theory, a well-established area of functional analysis. We also show how λδ presents better equational properties than other existing semantics and use our semantics to reason about a simplified ray tracing algorithm. Further, we relate our semantics to existing differentiable languages by providing translations to and from other existing differentiable semantic models. Finally, we provide a proof-of-concept implementation in PyTorch of the novel constructions in this paper.