  This work extends the theory of identifiability in supervised learning by
considering the consequences of having access to a distribution of tasks. In
such cases, we show that identifiability is achievable even in the case of
regression, extending prior work restricted to linear identifiability in the
single-task classification case. Furthermore, we show that the existence of a
task distribution which defines a conditional prior over latent factors reduces
the equivalence class for identifiability to permutations and scaling, a much
stronger and more useful result than linear identifiability. When we further
assume a causal structure over these tasks, our approach enables simple maximum
marginal likelihood optimization together with downstream applicability to
causal representation learning. Empirically, we validate that our model
outperforms more general unsupervised models in recovering canonical
representations for both synthetic and real-world molecular data.
