# On the Computation of Complex-valued Gradients with Application to Statistically Optimum Beamforming

arXiv: https://arxiv.org/abs/1701.00392 · HTML: https://mlsanity.com/paper/1701.00392/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jan 2017.

## Built on by (most cited first, 3 of 3)

- [Recurrent neural networks: vanishing and exploding gradients are not the end of the story](https://mlsanity.com/paper/2405.21064/) (2024)
- [Theory and Implementation of Complex-Valued Neural Networks](https://mlsanity.com/paper/2302.08286/) (2023)
- [Depth Completion with Multiple Balanced Bases and Confidence for Dense Monocular SLAM](https://mlsanity.com/paper/2309.04145/) (2023)

## Abstract (arXiv)

This report describes the computation of gradients by algorithmic differentiation for statistically optimum beamforming operations. Especially the derivation of complex-valued functions is a key component of this approach. Therefore the real-valued algorithmic differentiation is extended via the complex-valued chain rule. In addition to the basic mathematic operations the derivative of the eigenvalue problem with complex-valued eigenvectors is one of the key results of this report. The potential of this approach is shown with experimental results on the CHiME-3 challenge database. There, the beamforming task is used as a front-end for an ASR system. With the developed derivatives a joint optimization of a speech enhancement and speech recognition system w.r.t. the recognition optimization criterion is possible.