Diffusion-based inpainting of incomplete Euclidean distance matrices of trajectories generated by a fractional Brownian motion

Fractional Brownian trajectories (fBm) feature both randomness and strong scale-free correlations, challenging generative models to reproduce the intrinsic memory characterizing the underlying process. Here we test a diffusion probabilistic model on a specific dataset of corrupted images corresponding to incomplete Euclidean distance matrices of fBm at various memory exponents $H$. Our dataset implies uniqueness of the data imputation in the regime of low missing ratio, where the remaining partial graph is rigid, providing the ground truth for the inpainting. We find that the conditional diffusion generation stably reproduces the statistics of missing fBm-distributed distances for different values of $H$ exponent. Furthermore, while diffusion models have been recently shown to remember samples from the training database, we show that diffusion-based inpainting behaves qualitatively different from the database search with the increasing database size. Finally, we apply our fBm-trained diffusion model with $H=1/3$ for completion of chromosome distance matrices obtained in single-cell microscopy experiments, showing its superiority over the standard bioinformatics algorithms. Our source code is available on GitHub at https://github.com/alobashev/diffusion_fbm.