# Toward Complex-Valued Neural Networks for Waveform Generation

arXiv: https://arxiv.org/abs/2603.11589 · HTML: https://mlsanity.com/paper/2603.11589/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 36 of its 54 references that are in the map.
- First posted on arXiv 12 Mar 2026.

## Builds on (most cited first, 30 of 36)

- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [LibriTTS: A Corpus Derived from LibriSpeech for Text-to-Speech](https://mlsanity.com/paper/1904.02882/) (2019)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [UTMOS: UTokyo-SaruLab System for VoiceMOS Challenge 2022](https://mlsanity.com/paper/2204.02152/) (2022)
- [MelGAN: Generative Adversarial Networks for Conditional Waveform Synthesis](https://mlsanity.com/paper/1910.06711/) (2019)
- [Parallel WaveGAN: A fast waveform generation model based on generative adversarial networks with multi-resolution spectrogram](https://mlsanity.com/paper/1910.11480/) (2019)
- [BigVGAN: A Universal Neural Vocoder with Large-Scale Training](https://mlsanity.com/paper/2206.04658/) (2022)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Parallel WaveNet: Fast High-Fidelity Speech Synthesis](https://mlsanity.com/paper/1711.10433/) (2017)
- [Vocos: Closing the gap between time-domain and Fourier-based neural vocoders for high-quality audio synthesis](https://mlsanity.com/paper/2306.00814/) (2023)
- [Matcha-TTS: A fast TTS architecture with conditional flow matching](https://mlsanity.com/paper/2309.03199/) (2023)
- [Neural Networks Fail to Learn Periodic Functions and How to Fix It](https://mlsanity.com/paper/2006.08195/) (2020)
- [PriorGrad: Improving Conditional Denoising Diffusion Models with Data-Dependent Adaptive Prior](https://mlsanity.com/paper/2106.06406/) (2021)
- [UnivNet: A Neural Vocoder with Multi-Resolution Spectrogram Discriminators for High-Fidelity Waveform Generation](https://mlsanity.com/paper/2106.07889/) (2021)
- [Estimating or Propagating Gradients Through Stochastic Neurons](https://mlsanity.com/paper/1305.2982/) (2013)
- [WaveFlow: A Compact Flow-based Model for Raw Audio](https://mlsanity.com/paper/1912.01219/) (2019)
- [A Spectral Energy Distance for Parallel Speech Synthesis](https://mlsanity.com/paper/2008.01160/) (2020)
- [Chunked Autoregressive GAN for Conditional Waveform Synthesis](https://mlsanity.com/paper/2110.10139/) (2021)
- [iSTFTNet: Fast and Lightweight Mel-Spectrogram Vocoder Incorporating Inverse Short-Time Fourier Transform](https://mlsanity.com/paper/2203.02395/) (2022)
- [The universal approximation theorem for complex-valued neural networks](https://mlsanity.com/paper/2012.03351/) (2020)
- [Complex-Valued vs. Real-Valued Neural Networks for Classification Perspectives: An Example on Non-Circular Data](https://mlsanity.com/paper/2009.08340/) (2020)
- [PeriodWave: Multi-Period Flow Matching for High-Fidelity Waveform Generation](https://mlsanity.com/paper/2408.07547/) (2024)
- [Expediting TTS Synthesis with Adversarial Vocoding](https://mlsanity.com/paper/1904.07944/) (2019)
- [NanoFlow: Scalable Normalizing Flows with Sublinear Parameter Complexity](https://mlsanity.com/paper/2006.06280/) (2020)
- [RFWave: Multi-band Rectified Flow for Audio Waveform Reconstruction](https://mlsanity.com/paper/2403.05010/) (2024)
- [Generative adversarial network-based approach to signal reconstruction from magnitude spectrograms](https://mlsanity.com/paper/1804.02181/) (2018)
- [Optimal approximation using complex-valued neural networks](https://mlsanity.com/paper/2303.16813/) (2023)

## Abstract (arXiv)

Neural vocoders have recently advanced waveform generation, yielding natural and expressive audio. Among these approaches, iSTFT-based vocoders have recently gained attention. They predict a complex-valued spectrogram and then synthesize the waveform via iSTFT, thereby avoiding learned upsampling stages that can increase computational cost. However, current approaches use real-valued networks that process the real and imaginary parts independently. This separation limits their ability to capture the inherent structure of complex spectrograms. We present ComVo, a Complex-valued neural Vocoder whose generator and discriminator use native complex arithmetic. This enables an adversarial training framework that provides structured feedback in complex-valued representations. To guide phase transformations in a structured manner, we introduce phase quantization, which discretizes phase values and regularizes the training process. Finally, we propose a block-matrix computation scheme to improve training efficiency by reducing redundant operations. Experiments demonstrate that ComVo achieves higher synthesis quality than comparable real-valued baselines, and that its block-matrix scheme reduces training time by 25%. Audio samples and code are available at https://hs-oh-prml.github.io/ComVo/.