This paper explores the use of sparse Variational Autoencoders (VAE) for the analysis of distribution shifts in industrial datasets. To this end, several models are compared, in particular, we introduce the LassoVAE, a sparse model with a computationally efficient training. Comparisons are obtained thanks to an experimental protocol we designed that allows to generate synthetic data and different types of shifts with various parameters. New metrics are also introduced to evaluate the models' ability to retrieve the sources of shifts. Results show that sparse models are highly more efficient at recovering the true interactions between variables than a VAE with a dense decoder.