Tensor Network Moral Graph Recovery of Discrete Probability Distributions

Abstract: We present a method for recovering the moral graph of a causal DAG from a probability distribution over discrete variables, using fully connected tensor networks (FCTNs) with nuclear-norm-regularized bond corrections. Each bond matrix is parameterized as a baseline all-ones matrix plus a low-rank correction $C_{ij} = U_{ij}V_{ij}<sup>\top$, and the nuclear norm of the correction implemented via the variational Frobenius norm penalty on the factors drives unnecessary bonds to zero. We prove that under faithfulness, positivity, and a no-implicit-rerouting assumption on the local tensor architecture, \textbf{every} optimal FCTN with zero reconstruction error $\varepsilon = 0$ has effective graph exactly equal to the moral graph. For the approximate regime ($\varepsilon &gt; 0$), we provide explicit recovery bounds using the Fannes-Audenaert continuity of conditional mutual information, and derive a sufficient condition on the regularization parameter $β$. The effective graph is read directly from the optimized bond matrices.