  The theory of representation learning aims to build methods that provably
invert the data generating process with minimal domain knowledge or any source
of supervision. Most prior approaches require strong distributional assumptions
on the latent variables and weak supervision (auxiliary information such as
timestamps) to provide provable identification guarantees. In this work, we
show that if one has weak supervision from observations generated by sparse
perturbations of the latent variables--e.g. images in a reinforcement learning
environment where actions move individual sprites--identification is achievable
under unknown continuous latent distributions. We show that if the
perturbations are applied only on mutually exclusive blocks of latents, we
identify the latents up to those blocks. We also show that if these
perturbation blocks overlap, we identify latents up to the smallest blocks
shared across perturbations. Consequently, if there are blocks that intersect
in one latent variable only, then such latents are identified up to permutation
and scaling. We propose a natural estimation procedure based on this theory and
illustrate it on low-dimensional synthetic and image-based experiments.
