Backpropagation from KL Projections: Differential and Exact I-Projection Correspondences

We establish two correspondences between reverse-mode automatic differentiation (backpropagation at a given forward-pass point) and compositions of projection maps in Kullback--Leibler (KL) geometry. In both settings, message passing enforces agreement and factorization constraints through KL projections. In the first setting, backpropagation arises as the differential of a KL projection map on a lifted deterministic computation graph. In the second setting, on complete and decomposable sum--product networks, the same reverse-mode quantities coincide with exact probabilistic marginals and are realized by a KL I-projection. The distinction is that, in the first setting, projection induces structure, whereas, in the second, structure makes the projection exact. This pedagogical note highlights the relation among backpropagation, belief propagation, and KL projection algorithms and provides a perspective that unifies learning, sampling, and inference under a common geometric operator.