Iterative $\alpha$-(de)Blending: Learning a Deterministic Mapping Between Arbitrary Densities

We present a learning method that produces a mapping between arbitrary densities, such that random samples of a density can be mapped to random samples of another. In practice, our method is similar to deterministic diffusion processes where samples of the target density are blended with Gaussian noise. The originality of our approach is that, in contrast to several recent works, we do not rely on Langevin dynamics or score-matching concepts. We propose a simpler take on the topic, which is based solely on Bayes' theorem. By studying blended samples and their posteriors, we show that iteratively blending and deblending samples produces random paths between arbitrary densities. We prove that, for finite-variance densities, these paths converge towards a deterministic mapping that can be learnt with a neural network trained to deblend samples. Our method can thus be seen as a generalization of deterministic denoising diffusion where, instead of learning to denoise Gaussian noise, we learn to deblend arbitrary data.