  Motivated by distributed machine learning settings such as Federated
Learning, we consider the problem of fitting a statistical model across a
distributed collection of heterogeneous data sets whose similarity structure is
encoded by a graph topology. Precisely, we analyse the case where each node is
associated with fitting a sparse linear model, and edges join two nodes if the
difference of their solutions is also sparse. We propose a method based on
Basis Pursuit Denoising with a total variation penalty, and provide finite
sample guarantees for sub-Gaussian design matrices. Taking the root of the tree
as a reference node, we show that if the sparsity of the differences across
nodes is smaller than the sparsity at the root, then recovery is successful
with fewer samples than by solving the problems independently, or by using
methods that rely on a large overlap in the signal supports, such as the group
Lasso. We consider both the noiseless and noisy setting, and numerically
investigate the performance of distributed methods based on Distributed
Alternating Direction Methods of Multipliers (ADMM) and hyperspectral unmixing.
