mlsanityData to 7 Oct 2026

Quantitative approximation results for complex-valued neural networks

arXiv 2102.13092 ↗First posted 25 Feb 2021math.FA cs.LGcited by 5

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Abstract (arXiv)

Until recently, applications of neural networks in machine learning have almost exclusively relied on real-valued networks. It was recently observed, however, that complex-valued neural networks (CVNNs) exhibit superior performance in applications in which the input is naturally complex-valued, such as MRI fingerprinting. While the mathematical theory of real-valued networks has, by now, reached some level of maturity, this is far from true for complex-valued networks. In this paper, we analyze the expressivity of complex-valued networks by providing explicit quantitative error bounds for approximating Cn functions on compact subsets of mathbbCd by complex-valued neural networks that employ the modReLU activation function, given by σ(z) = ReLU(|z| - 1) , sgn (z), which is one of the most popular complex activation functions used in practice. We show that the derived approximation rates are optimal (up to log factors) in the class of modReLU networks with weights of moderate growth.

Key facts. Cited by 5 papers in the map (data through 7 Oct 2026). Builds on 17 of its 34 references that are in the map. First posted on arXiv 25 Feb 2021.

Builds on

17 of 34 references.

References · 17

  1. 2014Sequence to Sequence Learning with Neural Networkscited by 2,974
  2. 2016Error bounds for approximations with deep ReLU networkscited by 567
  3. 2017Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networkscited by 304
  4. 2015Unitary Evolution Recurrent Neural Networkscited by 291
  5. 2017Deep Complex Networkscited by 215
  6. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 176
  7. 2020Deep Network Approximation for Smooth Functionscited by 163
  8. 2019Deep Neural Network Approximation Theorycited by 92
  9. 2017Optimal Approximation with Sparsely Connected Deep Neural Networkscited by 90
  10. 2019The phase diagram of approximation rates for deep neural networkscited by 86
  11. 2016Depth-Width Tradeoffs in Approximating Natural Functions with Neural Networkscited by 58
  12. 2018Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equationscited by 44
5 more
  1. 2017Better than Real: Complex-valued Neural Nets for MRI Fingerprintingcited by 30
  2. 2020The universal approximation theorem for complex-valued neural networkscited by 28
  3. 2020Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditionscited by 25
  4. 2018Complex Gated Recurrent Neural Networkscited by 17
  5. 2019Uniform error estimates for artificial neural network approximations for heat equationscited by 11

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 1; 2023: 2; 2024: 0; 2025: 0; 2026: 2.

5 papers.

  1. 2023Deep Operator Network Approximation Rates for Lipschitz Operatorscited by 24
  2. 2023Universal approximation with complex-valued deep narrow neural networkscited by 3
  3. 2022Towards Antisymmetric Neural Ansatz Separationcited by 2
  4. 2026PHALAR: Phasors for Learned Musical Audio Representationscited by 1
  5. 2026Theory Guided and Interpretable Neural Operator Design for Partial Differential Equation Learningcited by 0