  Identifiability is a desirable property of a statistical model: it implies
that the true model parameters may be estimated to any desired precision, given
sufficient computational resources and data. We study identifiability in the
context of representation learning: discovering nonlinear data representations
that are optimal with respect to some downstream task. When parameterized as
deep neural networks, such representation functions typically lack
identifiability in parameter space, because they are overparameterized by
design. In this paper, building on recent advances in nonlinear ICA, we aim to
rehabilitate identifiability by showing that a large family of discriminative
models are in fact identifiable in function space, up to a linear
indeterminacy. Many models for representation learning in a wide variety of
domains have been identifiable in this sense, including text, images and audio,
state-of-the-art at time of publication. We derive sufficient conditions for
linear identifiability and provide empirical support for the result on both
simulated and real-world data.
