  Fourier analysis has been an instrumental tool in the development of signal
processing. This leads us to wonder whether this framework could similarly
benefit generative modelling. In this paper, we explore this question through
the scope of time series diffusion models. More specifically, we analyze
whether representing time series in the frequency domain is a useful inductive
bias for score-based diffusion models. By starting from the canonical SDE
formulation of diffusion in the time domain, we show that a dual diffusion
process occurs in the frequency domain with an important nuance: Brownian
motions are replaced by what we call mirrored Brownian motions, characterized
by mirror symmetries among their components. Building on this insight, we show
how to adapt the denoising score matching approach to implement diffusion
models in the frequency domain. This results in frequency diffusion models,
which we compare to canonical time diffusion models. Our empirical evaluation
on real-world datasets, covering various domains like healthcare and finance,
shows that frequency diffusion models better capture the training distribution
than time diffusion models. We explain this observation by showing that time
series from these datasets tend to be more localized in the frequency domain
than in the time domain, which makes them easier to model in the former case.
All our observations point towards impactful synergies between Fourier analysis
and diffusion models.
