Parameter-efficient finetuning methods based on spectral decomposition have enabled progress in Continual Learning. In this paper we introduce TailLoR, which utilizes the singular bases U and V of the pre-trained weights as a fixed reference frame to learn a low-rank update applied to the singular value matrix. A soft spectral penalty discourages updates aligned with dominant singular directions, reducing interference while routing fine-grained adaptation into the highly flexible, long-tail spectral coordinates.

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