Efficient Modular Learning through Naive LoRA Summation: Leveraging Orthogonality in High-Dimensional Models

Abstract: Recent advances in LLMs are driven by scale, while parameter-efficient fine-tuning (PEFT) enables updating only a small fraction of parameters. Low-Rank Adaptation (LoRA) stores parameter deltas as the product of two small matrices, which makes them natural building blocks that can be composed. Motivated by the superposition principle, we hypothesize that independently trained LoRA modules on disjoint domains are approximately orthogonal and can be combined by simple addition. Using GPT-2 Small (117M) with LoRA rank 4 and alpha=64, we train adapters for three QA domains (math, medicine, finance). In pairwise tests, adding Math+Medicine adapters improves perplexity by -9.10% relative to merged-data fine-tuning, while Math+Finance and Finance+Medicine change by +4.54% and +27.56%, respectively. Across combinations, the RMS cosine similarity between LoRA deltas correlates positively and approximately linearly with the change in perplexity. Naive summation requires no additional training, can be applied in seconds, and achieves performance comparable to models trained on merged data, while clarifying when interference appears in higher-order compositions.