The Complex Gradient Operator and the CR-Calculus ECE275A - Lecture Supplement - Fall 2005

By Kenneth Kreutz-Delgado. 2009

497 citations recorded by the source corpus.

A thorough discussion and development of the calculus of real-valued functions of complex-valued vectors is given using the framework of the Wirtinger Calculus. The presented material is suitable for exposition in an introductory Electrical Engineering graduate level course on the use of complex gradients and complex Hessian matrices, and has been successfully used in teaching at UC San Diego. Going beyond the commonly encountered treatments of the first-order complex vector calculus, second-order considerations are examined in some detail filling a gap in the pedagogic literature.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ccfcf7025de0c852213a99f2ffbe108e823552aa

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Citations in the corpus

Complex Elliptically Symmetric Distributions: Survey, New Results and Applications (2012, background): “, are also called the -derivative and the conjugate -derivative, respectively, and the differential calculus based on these operators is called the -calculus or Wirtinger calculus [107]; we refer the reader to [13], [26], [27], [53], [56], [66], [82] and [1] for a detailed account of complex partial”
Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution (2017, background): “There is a one-to-one mapping connecting the Wirtinger derivatives and the conventional derivatives [69]: [ x y ] = J−1 [ z z ] , (280a)”
A Geometric Analysis of Phase Retrieval (2016, unknown): “The first results of this nature were obtained using semidefinite programming (SDP) relaxations [28–30,32,102,104].”
Better than real: Complex-valued neural nets for MRI fingerprinting (2017, methodology): “We leverage Wirtinger calculus, or CR calculus, [11, 10] to do gradient descent on functions that are not complex differentiable as long as they are differentiable with respect to their real and imaginary components.”
Adaptation, Learning, and Optimization over Networks (2014, background): “For these cases, it is sufficient to rely on a different notion of differentiation, which we proceed to motivate following [3, 47, 107, 111, 116, 198, 207, 219, 252].”
Experimental robustness of Fourier Ptychography phase retrieval algorithms (2015, background): “From [17,30], the definition for the Hessian of a real-value function with multiple complex variables is a 2n(2) × 2n(2) matrix and can be expressed as”
New Era for Robust Speech Recognition (2017, methodology): “An example of a wideband approach is that of [29], which combines ideas from [10, 28, 53].”
Complex-Valued Matrix Differentiation: Techniques and Key Results (2007, background): “The problem at hand has been treated for real-valued matrix variables in [2], [1], [3], [4], [5].”