Wirtinger calculus

It’s not complicated / It’s complex

May 8, 2019 — September 7, 2021

Figure 1

How do you differentiate real-valued functions of complex arguments? Wirtinger calculus. This is a convenient hack that happens to work well for most practical signal processing over the complex field, especially in optimisation. It arises naturally in, for example, phase retrieval (H. Zhang and Liang 2016; Candes, Li, and Soltanolkotabi 2015; Chen and Candès 2015; Seuret and Gouaisbaut 2013). Because of its area of popularity, this will almost surely arise in combination also with matrix calculus.

There are various write-ups of Wirtinger calculus; this will be a literature review of those because they are of highly variable quality and I want to ensure I understand what I’m playing with here.

Bouboulis (2010) has a punchy intro as part of a paper which takes Wirtinger derivatives inside inner products:

Wirtinger’s calculus has become very popular in the signal processing community mainly in the context of complex adaptive filtering, as a means of computing, in an elegant way, gradients of real valued cost functions defined on complex domains ( $^{} $ ). Such functions, obviously, are not holomorphic and therefore the complex derivative cannot be used. Instead, if we consider that the cost function is defined on a Euclidean domain with a double dimensionality ($^{2} $), then the real derivatives may be employed. The price of this approach is that the computations become cumbersome and tedious. Wirtinger’s calculus provides an alternative equivalent formulation, that is based on simple rules and principles and which bears a great resemblance to the rules of the standard complex derivative… A common misconception …is that Wirtinger’s calculus uses an alternative definition of derivatives and therefore results in different gradient rules in minimization problems. We should emphasize that the theoretical foundation of Wirtinger’s calculus is the common definition of the real derivative. However, it turns out that when the complex structure is taken into account, the real derivatives may be described using an equivalent and more elegant formulation which bears a surprising resemblance with the complex derivative. Therefore, simple rules may be derived and the computations of the gradients, which may become tedious if the double dimensional space $^{2} $, is considered, are simplified.

The extension to Hilbert space operations is nifty.

Other favoured resources:

1 References

Adali, Schreier, and Scharf. 2011. Complex-Valued Signal Processing: The Proper Way to Deal With Impropriety.” IEEE Transactions on Signal Processing.
Bouboulis. 2010. Wirtinger’s Calculus in General Hilbert Spaces.” arXiv:1005.5170 [Cs, Math].
Candes, Li, and Soltanolkotabi. 2015. Phase Retrieval via Wirtinger Flow: Theory and Algorithms.” IEEE Transactions on Information Theory.
Caracalla, and Roebel. 2017. Gradient Conversion Between Time and Frequency Domains Using Wirtinger Calculus.” In International Conference on Digital Audio Effects (DAFx-17), Edinburgh, UK, September 5–9, 2017.
Chen, and Candès. 2015. Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems.” In Advances in Neural Information Processing Systems 28.
Fan, Liao, and Liu. 2016. An Overview of the Estimation of Large Covariance and Precision Matrices.” The Econometrics Journal.
Fischer. 2005. Appendix A: Wirtinger Calculus.” In Precoding and Signal Shaping for Digital Transmission.
Hjørungnes. 2011. Complex-Valued Matrix Derivatives: With Applications in Signal Processing and Communications.
Hunger. 2007. “An Introduction to Complex Differentials and Complex Differentiability.”
Schreier, and Scharf. 2010. Statistical Signal Processing of Complex-Valued Data: The Theory of Improper and Noncircular Signals.
Seuret, and Gouaisbaut. 2013. Wirtinger-Based Integral Inequality: Application to Time-Delay Systems.” Automatica.
Zhang, Huishuai, and Liang. 2016. Reshaped Wirtinger Flow for Solving Quadratic System of Equations.” In Advances in Neural Information Processing Systems 29.
Zhang, T., and Zou. 2014. Sparse Precision Matrix Estimation via Lasso Penalized D-Trace Loss.” Biometrika.