Structurally Separated DAG Learning with Multi-Scale Normalized Closure

Shuai Feng, Hao Yan

Continuous DAG learning commonly enforces acyclicity through scalar functions of a weighted adjacency that aggregate evidence from weighted closed walks. We add a complementary layer: matrix-valued closure logits---explicit multi-hop structural relation scores for each ordered pair---supervised alongside data fit, sparsity, and a DAGMA-style barrier on the weighted adjacency. The distinction is not scalarization of the overall objective but how path evidence is represented and exposed before aggregation. SMNC (scale-normalized multi-scale closure) builds these scores on a structural gate by repeated squaring in log space and divides each scale by its path-length budget, counteracting product-style attenuation along long cycles; diagonal entries support node-wise cycle supervision and off-diagonals optional pairwise reachability cues. In the linear model, a bounded structure--magnitude factorization ties strong effective edges to strong gates; a nonlinear induced gate offers another construction with the same intent. For sufficiently strong cycles, normalized logits admit a length-independent lower bound; under a unique-active-cycle assumption, the hard final-scale readout yields…