# Tensor Network Moral Graph Recovery of Discrete Probability Distributions

arXiv: https://arxiv.org/abs/2609.09258 · HTML: https://mlsanity.com/paper/2609.09258/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 19 references that are in the map.
- First posted on arXiv 8 Sep 2026.

## Builds on (most cited first, 1 of 1)

- [Causal Inference and Causal Explanation with Background Knowledge](https://mlsanity.com/paper/1302.4972/) (2013)

## Abstract (arXiv)

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 Cij = UijVij^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, every optimal FCTN with zero reconstruction error varepsilon = 0 has effective graph exactly equal to the moral graph. For the approximate regime (varepsilon > 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.