From Coverage Rules to Unbalanced Optimal Transport: A Mathematical Analysis of Bidirectional Topic Alignment

Source abstract

Text that accumulates over time does not only drift: topics are born, they die, they fragment into competing sub-narratives (splits), and they fuse into broader frames (merges). Our companion system paper, BERTilda, detects these events with transparent similarity and flow rules and validates them against human annotators; the present article is its mathematical companion. We show that the forward (outflow) and backward (inflow) coverages used by BERTilda are two semi-couplings between the topics of adjacent windows, both one-sided limits of a single unbalanced optimal-transport problem. The transport view separates the modeling layer from the readout layer: the plan is governed by three interpretable transport parameters, a blur and two marginal fidelities, and by a cost matrix fixed by the embedding geometry, while lifecycle labels are read from two explicit decision levels for survival and dominance. Because the plan is a stable function of that geometry and the topic centroids concentrate around their latent positions, we obtain finite-sample guarantees that account for the noise in the centroids themselves, not just in document attribution. We prove that the recovered transport structure converges to the latent geometry and that labels are recovered once their margin exceeds the plan’s perturbation radius; a lower bound for an edge-decision subproblem shows that the resulting inverse-square dependence on the decision margin cannot be improved in general. The theory also offers an explanation of BERTilda’s empirical pattern: continuation and disappearance are easy, while split and merge have the smallest attainable margins.