A tutorial on automatic differentiation with complex numbers
<|reference_start|>A tutorial on automatic differentiation with complex numbers: Automatic differentiation is everywhere, but there exists only minimal documentation of how it works in complex arithmetic beyond stating "derivatives in $\mathbb{C}^d$" $\cong$ "derivatives in $\mathbb{R}^{2d}$" and, at best, shallow references to Wirtinger calculus. Unfortunately, the equivalence $\mathbb{C}^d \cong \mathbb{R}^{2d}$ becomes insufficient as soon as we need to derive custom gradient rules, e.g., to avoid differentiating "through" expensive linear algebra functions or differential equation simulators. To combat such a lack of documentation, this article surveys forward- and reverse-mode automatic differentiation with complex numbers, covering topics such as Wirtinger derivatives, a modified chain rule, and different gradient conventions while explicitly avoiding holomorphicity and the Cauchy--Riemann equations (which would be far too restrictive). To be precise, we will derive, explain, and implement a complex version of Jacobian-vector and vector-Jacobian products almost entirely with linear algebra without relying on complex analysis or differential geometry. This tutorial is a call to action, for users and developers alike, to take complex values seriously when implementing custom gradient propagation rules -- the manuscript explains how.<|reference_end|>
@article{krämer2024a,
    title={A tutorial on automatic differentiation with complex numbers},
    author={Nicholas Kr"amer},
    journal={arXiv preprint arXiv:2409.06752},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.06752},
    primaryClass={cs.MS cs.LG cs.NA math.NA}
}