When Does Subspace Direction Matter for LoRA? Regime Analysis of the Magnitude Principle in Few-Shot Adaptation

Low-Rank Adaptation (LoRA) constrains weight updates to a low-rank subspace, making the choice of that subspace central to adaptation. We show that the value of LoRA initialization direction is regime-dependent. It helps when the representation dimension is large relative to the labeled set size and the estimated activation subspace is aligned with task structure, and is otherwise inconsequential. We formalize this view with a spiked-covariance Davis--Kahan argument and a first-step LoRA analysis showing that the initial learning signal flows through projected target-token features. From this analysis we derive an alignment coefficient $\rho$, computable from a single forward pass, that closely tracks the empirical gain from direction-aware initialization across tasks. Motivated by this view, we propose PivotLoRA, an initialization that estimates LoRA adapters from target-token PCA and applies a single output-preserving orthogonal pivot after early training drift. Across $30$ BERT-base few-shot settings and $13$ compared methods, PivotLoRA achieves the best average rank and improves over vanilla LoRA on nearly every setting, with the largest gains concentrated on high-alignment tasks. Decoder-scale classification on LLaMA-2-7B and Llama-3.1-8B reproduces the low-shot pattern, while full-data and generation controls confirm the predicted regime specificity.