  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.

Audio and Speech Processing;Machine Learning