  How can intelligent agents solve a diverse set of tasks in a data-efficient
manner? The disentangled representation learning approach posits that such an
agent would benefit from separating out (disentangling) the underlying
structure of the world into disjoint parts of its representation. However,
there is no generally agreed-upon definition of disentangling, not least
because it is unclear how to formalise the notion of world structure beyond toy
datasets with a known ground truth generative process. Here we propose that a
principled solution to characterising disentangled representations can be found
by focusing on the transformation properties of the world. In particular, we
suggest that those transformations that change only some properties of the
underlying world state, while leaving all other properties invariant, are what
gives exploitable structure to any kind of data. Similar ideas have already
been successfully applied in physics, where the study of symmetry
transformations has revolutionised the understanding of the world structure. By
connecting symmetry transformations to vector representations using the
formalism of group and representation theory we arrive at the first formal
definition of disentangled representations. Our new definition is in agreement
with many of the current intuitions about disentangling, while also providing
principled resolutions to a number of previous points of contention. While this
work focuses on formally defining disentangling - as opposed to solving the
learning problem - we believe that the shift in perspective to studying data
transformations can stimulate the development of better representation learning
algorithms.
