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

TL;DR

This paper develops methods for computing complex-valued gradients using algorithmic differentiation, enabling joint optimization of speech enhancement and recognition systems with statistically optimal beamforming.

Contribution

It extends real-valued algorithmic differentiation to complex functions, including derivatives of eigenvalue problems with complex eigenvectors, for beamforming applications.

Findings

Successful application to CHiME-3 database

Joint optimization improves speech recognition performance

Demonstrates the effectiveness of complex gradient computation in beamforming

Abstract

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.