  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.
