Row Amplitude as an Optimization Mechanism in Transport-Based Learning

A transport target determines both a sample's assignment and its weight in learning. We study the latter—its row amplitude—as a downstream optimization variable, separating its effect from changes in conditional assignment. For losses linear in a stopped target, Row-Mass Tempering (RMT) isolates amplitude while preserving assignments, the set of nonzero rows, and total mass. An exact mass–force covariance explains the resulting gradient correction and when scalar rescaling cannot reproduce it. This motivates Row-Amplitude Audit (RAA), which measures the actual optimizer effect and selects a coefficient from initial training observations for prospective validation. Frozen rules meet the declared update-deviation criterion in all three P${}^2$OT seeds and all five JUMBOT seeds. Separate fixed-intervention studies establish task value: development-selected RMT improves PROTOCOL/Hdigit clustering accuracy by 5.09 percentage points and adjusted Rand index by 6.56 points over 10 paired seeds; equalizing active row amplitudes improves these metrics by 5.60 and 6.53 points in P${}^2$OT/CIFAR100-LT over five paired seeds, with lower class-balanced accuracy and purity. JUMBOT shows the boundary: successful optimizer control yields no observed accuracy gain over the fixed reference. The contribution is a systematic account of an amplitude channel that can be isolated, explained, measured, and controlled, with optimizer controllability and downstream utility evaluated as distinct objectives.