# Adversarial Audio Synthesis with Complex-valued Polynomial Networks

arXiv: https://arxiv.org/abs/2206.06811 · HTML: https://mlsanity.com/paper/2206.06811/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 73 references that are in the map.
- First posted on arXiv 14 Jun 2022.

## Builds on (most cited first, 30 of 47)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Large Scale GAN Training for High Fidelity Natural Image Synthesis](https://mlsanity.com/paper/1809.11096/) (2018)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Non-local Neural Networks](https://mlsanity.com/paper/1711.07971/) (2017)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Complex Embeddings for Simple Link Prediction](https://mlsanity.com/paper/1606.06357/) (2016)
- [Natural TTS Synthesis by Conditioning WaveNet on Mel Spectrogram Predictions](https://mlsanity.com/paper/1712.05884/) (2017)
- [Speech Commands: A Dataset for Limited-Vocabulary Speech Recognition](https://mlsanity.com/paper/1804.03209/) (2018)
- [Tacotron: Towards End-to-End Speech Synthesis](https://mlsanity.com/paper/1703.10135/) (2017)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [cGANs with Projection Discriminator](https://mlsanity.com/paper/1802.05637/) (2018)
- [Neural Audio Synthesis of Musical Notes with WaveNet Autoencoders](https://mlsanity.com/paper/1704.01279/) (2017)
- [Adversarial Audio Synthesis](https://mlsanity.com/paper/1802.04208/) (2018)
- [Efficient Neural Audio Synthesis](https://mlsanity.com/paper/1802.08435/) (2018)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [It's Raw! Audio Generation with State-Space Models](https://mlsanity.com/paper/2202.09729/) (2022)
- [GANSynth: Adversarial Neural Audio Synthesis](https://mlsanity.com/paper/1902.08710/) (2019)
- [SampleRNN: An Unconditional End-to-End Neural Audio Generation Model](https://mlsanity.com/paper/1612.07837/) (2016)
- [DeLiGAN : Generative Adversarial Networks for Diverse and Limited Data](https://mlsanity.com/paper/1706.02071/) (2017)
- [Phase-aware Speech Enhancement with Deep Complex U-Net](https://mlsanity.com/paper/1903.03107/) (2019)
- [Deep Polynomial Neural Networks](https://mlsanity.com/paper/2006.13026/) (2020)
- [On GANs and GMMs](https://mlsanity.com/paper/1805.12462/) (2018)

## Built on by (most cited first, 3 of 3)

- [Polynomial Neural Fields for Subband Decomposition and Manipulation](https://mlsanity.com/paper/2302.04862/) (2023)
- [Hadamard product in deep learning: Introduction, Advances and Challenges](https://mlsanity.com/paper/2504.13112/) (2025)
- [Polynomial-Augmented Neural Networks (PANNs) with Weak Orthogonality Constraints for Enhanced Function and PDE Approximation](https://mlsanity.com/paper/2406.02336/) (2024)

## Abstract (arXiv)

Time-frequency (TF) representations in audio synthesis have been increasingly modeled with real-valued networks. However, overlooking the complex-valued nature of TF representations can result in suboptimal performance and require additional modules (e.g., for modeling the phase). To this end, we introduce complex-valued polynomial networks, called APOLLO, that integrate such complex-valued representations in a natural way. Concretely, APOLLO captures high-order correlations of the input elements using high-order tensors as scaling parameters. By leveraging standard tensor decompositions, we derive different architectures and enable modeling richer correlations. We outline such architectures and showcase their performance in audio generation across four benchmarks. As a highlight, APOLLO results in 17.5% improvement over adversarial methods and 8.2% over the state-of-the-art diffusion models on SC09 dataset in audio generation. Our models can encourage the systematic design of other efficient architectures on the complex field.