Polynomial Expansion Rank Adaptation: Enhancing Low-Rank Fine-Tuning with High-Order Interactions

Low-rank adaptation (LoRA) is a widely used strategy for efficient fine-tuning of large language models (LLMs), but its strictly linear structure fundamentally limits expressive capacity. The bilinear formulation of weight updates captures only first-order dependencies between low-rank factors, restricting the modeling of nonlinear and higher-order parameter interactions. In this paper, we propose Polynomial Expansion Rank Adaptation (PERA), a novel method that introduces structured polynomial expansion directly into the low-rank factor space. By expanding each low-rank factor to synthesize high-order interaction terms before composition, PERA transforms the adaptation space into a polynomial manifold capable of modeling richer nonlinear coupling without increasing rank or inference cost. We provide theoretical analysis demonstrating that PERA offers enhanced expressive capacity and more effective feature utilization compare to existing linear adaptation approaches. Empirically, PERA consistently outperforms state-of-the-art methods across diverse benchmarks. Notably, our experiments show that incorporating high-order nonlinear components particularly square terms is crucial for enhancing expressive capacity and maintaining strong and robust performance under various rank settings. Our code is available at https://github.com/zhangwenhao6/PERA