  This paper proposes a general enhancement to the Normalizing Flows (NF) used
in neural vocoding. As a case study, we improve expressive speech vocoding with
a revamped Parallel Wavenet (PW). Specifically, we propose to extend the affine
transformation of PW to the more expressive invertible non-affine function. The
greater expressiveness of the improved PW leads to better-perceived signal
quality and naturalness in the waveform reconstruction and text-to-speech (TTS)
tasks. We evaluate the model across different speaking styles on a
multi-speaker, multi-lingual dataset. In the waveform reconstruction task, the
proposed model closes the naturalness and signal quality gap from the original
PW to recordings by $10\%$, and from other state-of-the-art neural vocoding
systems by more than $60\%$. We also demonstrate improvements in objective
metrics on the evaluation test set with L2 Spectral Distance and Cross-Entropy
reduced by $3\%$ and $6\unicode{x2030}$ comparing to the affine PW.
Furthermore, we extend the probability density distillation procedure proposed
by the original PW paper, so that it works with any non-affine invertible and
differentiable function.
