  Humans have a remarkable ability to disentangle complex sensory inputs (e.g.,
image, text) into simple factors of variation (e.g., shape, color) without much
supervision. This ability has inspired many works that attempt to solve the
following question: how do we invert the data generation process to extract
those factors with minimal or no supervision? Several works in the literature
on non-linear independent component analysis have established this negative
result; without some knowledge of the data generation process or appropriate
inductive biases, it is impossible to perform this inversion. In recent years,
a lot of progress has been made on disentanglement under structural
assumptions, e.g., when we have access to auxiliary information that makes the
factors of variation conditionally independent. However, existing work requires
a lot of auxiliary information, e.g., in supervised classification, it
prescribes that the number of label classes should be at least equal to the
total dimension of all factors of variation. In this work, we depart from these
assumptions and ask: a) How can we get disentanglement when the auxiliary
information does not provide conditional independence over the factors of
variation? b) Can we reduce the amount of auxiliary information required for
disentanglement? For a class of models where auxiliary information does not
ensure conditional independence, we show theoretically and experimentally that
disentanglement (to a large extent) is possible even when the auxiliary
information dimension is much less than the dimension of the true latent
representation.
