  Temporal data such as time series can be viewed as discretized measurements
of the underlying function. To build a generative model for such data we have
to model the stochastic process that governs it. We propose a solution by
defining the denoising diffusion model in the function space which also allows
us to naturally handle irregularly-sampled observations. The forward process
gradually adds noise to functions, preserving their continuity, while the
learned reverse process removes the noise and returns functions as new samples.
To this end, we define suitable noise sources and introduce novel denoising and
score-matching models. We show how our method can be used for multivariate
probabilistic forecasting and imputation, and how our model can be interpreted
as a neural process.
