A COORDINATE VIEW OF LORA: FIXED BASES AND LEARNED COMBINATIONS

Storing fewer parameters per task makes it less costly to adapt one model to many tasks. Understanding those parameters could also help us identify and modify the changes that affect task performance. Methods with fixed factors already reduce task-specific parameters, but their performance depends jointly on the basis and its learned coefficients. We separate basis information, scale, and coefficient budgets, then study the effects of changing the coefficients. The coordinate view $\Delta W=ANB$, with shared bases and task-specific coordinates $N$, provides a common framework for analyzing basis information, connectivity, and task directions. Normalized random bases closely approach pretrained-SVD performance, while learning-rate calibration substantially recovers raw Gaussian performance. At comparable coordinate budgets, wider sparse systems can outperform narrower dense ones, with connectivity effects also depending on update scale. Scaling task residuals forward or backward produces structured changes in accuracy; a single few-shot gradient also identifies beneficial directions from weak starting states. Across six tasks, the largest 10% of a 64-example direction's coordinates retain approximately 93% of its accuracy gain from zero and randomized starts; relocating the same values largely removes the benefit. These findings connect compact task-specific storage with interpretable, direction- and location-dependent changes in task performance.