SaMA: Morpho Adaptation via Asymmetric Expansion of Kronecker Product

Low-rank adaptation (LoRA) has attracted significant attention in the parameter-efficient fine-tuning (PEFT) landscape. However, its design suffers from two fundamental limitations. First, its invariance manifold induces flat directions in the loss landscape, creating an optimization bottleneck for adaptive optimizers. Second, its effective rank scales only linearly with the parameter budget. In this paper, we propose Scalable Morpho Adaptation (SaMA), a PEFT method grounded in a principled generalization of the Kronecker product. By relaxing the shared-block constraint in the perfect-shuffle factorization, SaMA spans from low to full rank and achieves effective rank that scales \quadratically in parameters. Furthermore, we show that its invariance manifold collapses to a diagonal group, making it structurally more optimizer-friendly. Empirically, SaMA consistently outperforms strong PEFT baselines on commonsense and arithmetic reasoning benchmarks while being more parameter-efficient.