  Principal Component Analysis (PCA) and its exponential family extensions have
three components: observations, latents and parameters of a linear
transformation. We consider a generalised setting where the canonical
parameters of the exponential family are a nonlinear transformation of the
latents. We show explicit relationships between particular neural network
architectures and the corresponding statistical models. We find that deep
equilibrium models -- a recently introduced class of implicit neural networks
-- solve maximum a-posteriori (MAP) estimates for the latents and parameters of
the transformation. Our analysis provides a systematic way to relate activation
functions, dropout, and layer structure, to statistical assumptions about the
observations, thus providing foundational principles for unsupervised DEQs. For
hierarchical latents, individual neurons can be interpreted as nodes in a deep
graphical model. Our DEQ feature maps are end-to-end differentiable, enabling
fine-tuning for downstream tasks.
