  Iterative refinement -- start with a random guess, then iteratively improve
the guess -- is a useful paradigm for representation learning because it offers
a way to break symmetries among equally plausible explanations for the data.
This property enables the application of such methods to infer representations
of sets of entities, such as objects in physical scenes, structurally
resembling clustering algorithms in latent space. However, most prior works
differentiate through the unrolled refinement process, which can make
optimization challenging. We observe that such methods can be made
differentiable by means of the implicit function theorem, and develop an
implicit differentiation approach that improves the stability and tractability
of training by decoupling the forward and backward passes. This connection
enables us to apply advances in optimizing implicit layers to not only improve
the optimization of the slot attention module in SLATE, a state-of-the-art
method for learning entity representations, but do so with constant space and
time complexity in backpropagation and only one additional line of code.
