  Normalising flows offer a flexible way of modelling continuous probability
distributions. We consider expressiveness, fast inversion and exact Jacobian
determinant as three desirable properties a normalising flow should possess.
However, few flow models have been able to strike a good balance among all
these properties. Realising that the integral of a convex sum of sinusoidal
functions squared leads to a bijective residual transformation, we propose
Sinusoidal Flow, a new type of normalising flows that inherits the expressive
power and triangular Jacobian from fully autoregressive flows while guaranteed
by Banach fixed-point theorem to remain fast invertible and thereby obviate the
need for sequential inversion typically required in fully autoregressive flows.
Experiments show that our Sinusoidal Flow is not only able to model complex
distributions, but can also be reliably inverted to generate realistic-looking
samples even with many layers of transformations stacked.
