RETHINKING SUBSPACE SIMILARITY IN LOW-RANK ADAPTATION: WHAT GOVERNS LOCAL UPDATES?

Low-rank adaptation is usually understood as optimization under a low-rank constraint, but its update map is not a projection: it weights a gradient by the spectrum the factors were built from. We reframe it as optimization under a spectrum-weighted map, and identify spectral allocation—how the spectrum is distributed across the two factors—as the variable governing early training, not the rank or the selected subspace. We test this reframing with paired upstream updates, shared-gradient probes, and natural-magnitude component interventions. A modified initialization can preserve its selected subspaces almost exactly while changing the weighted update map appreciably—a divergence we call the subspace–update gap. The distinction reaches training: two factorizations of one weight with identical initial effective weight, identical selected subspaces, and a shared learning rate produce ten-step updates lying 77.5°–88.0° apart, so spectral weighting changes the direction of adaptation, not only its speed. The gap is not an artifact of one spectral choice: cross-subspace coupling reproduces the response under principal-component selection but not under minor-component selection, where changes within the selected block dominate. The criterion is measurable with the same interventions and applies to any modification of the reference weights, including quantization and pruning.