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This work introduces a novel principle for disentanglement we call mechanismsparsity regularization, which applies when the latent factors of interestdepend sparsely on observed auxiliary variables and/or past latent factors. Wepropose a representation learning method that induces disentanglement bysimultaneously learning the latent factors and the sparse causal graphicalmodel that explains them. We develop a nonparametric identifiability theorythat formalizes this principle and shows that the latent factors can berecovered by regularizing the learned causal graph to be sparse. Moreprecisely, we show identifiablity up to a novel equivalence relation we call"consistency", which allows some latent factors to remain entangled (hence theterm partial disentanglement). To describe the structure of this entanglement,we introduce the notions of entanglement graphs and graph preserving functions.We further provide a graphical criterion which guarantees completedisentanglement, that is identifiability up to permutations and element-wisetransformations. We demonstrate the scope of the mechanism sparsity principleas well as the assumptions it relies on with several worked out examples. Forinstance, the framework shows how one can leverage multi-node interventionswith unknown targets on the latent factors to disentangle them. We further drawconnections between our nonparametric results and the now popular exponentialfamily assumption. Lastly, we propose an estimation procedure based onvariational autoencoders and a sparsity constraint and demonstrate it onvarious synthetic datasets. This work is meant to be a significantly extendedversion of Lachapelle et al. (2022).

Nonparametric Partial Disentanglement via Mechanism Sparsity: Sparse Actions, Interventions and Sparse Temporal Dependencies