MuLoRA: Spectrally Balanced Low-Rank Adaptation for Continual Learning

Low-rank adaptation (LoRA) provides a parameter-efficient approach to continual learning, but its nominal rank can conceal a loss of effective adaptation capacity. We identify spectral plasticity collapse: during sequential adaptation, update energy becomes concentrated in a small subset of singular modes, leaving much of the available low-rank space underutilized. This exposes a limitation of interference avoidance alone: protecting historical representations does not ensure that the remaining adaptation capacity is responsive to new tasks or effectively utilized. To address this problem, we propose textttMuLoRA, which jointly controls capacity allocation and utilization. First, historical whitening identifies input directions with strong current-task response relative to accumulated historical response, yielding a task-adaptive basis that remains fixed during training. Second, approximate polar orthogonalization of momentum updates reduces spectral concentration within theselected space. An orthonormal basis connects these mechanisms by transferring the factor-update spectrum exactly tothe induced weight update. We establish a max--min characterization of exact subspace selection and derive cumulative spectral bounds under controlled cross-step anisotropy. Across five class-incremental benchmarks and eight incremental settings, textttMuLoRA achieves the highest mean accuracy in 15 of 16 reported metrics.