Real Gâteaux and Wirtinger gradients for real-valued optimisation on complex Hilbert spaces
Year: 2026
Type: preprint
Abstract: Many problems in quantum physics, signal processing, phase retrieval, machine learning and other topics require the minimisation of real-valued functionals on complex Hilbert spaces. Since non-constant real-valued functionals on complex Hilbert spaces cannot be complex Gâteaux differentiable, gradient descent methods require alternative notions of derivative. Two commonly used approaches are given by the real Gâteaux derivative and the Wirtinger derivative. In this note, we analyse both strategies in the general setting of (infinite-dimensional) complex Hilbert spaces and show that they lead to the same notion of critical point and give rise to identical gradients, and hence identical steepest descent directions, with respect to any equivalent inner product.
Source: HAL
Author: Heid, Pascal
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7167872853