# SOS-LoRA: Static Orthogonal-Subspace Low-Rank Adaptation with Fixed Multi-Scale Scaling

arXiv: https://arxiv.org/abs/2607.16252 · HTML: https://mlsanity.com/paper/2607.16252/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 41 references that are in the map.
- First posted on arXiv 27 Jun 2026.

## Builds on (most cited first, 27 of 27)

- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [BoolQ: Exploring the Surprising Difficulty of Natural Yes/No Questions](https://mlsanity.com/paper/1905.10044/) (2019)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [Parameter-Efficient Fine-Tuning for Large Models: A Comprehensive Survey](https://mlsanity.com/paper/2403.14608/) (2024)
- [LoRA+: Efficient Low Rank Adaptation of Large Models](https://mlsanity.com/paper/2402.12354/) (2024)
- [AdaLoRA: Adaptive Budget Allocation for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2303.10512/) (2023)
- [Orthogonal Subspace Learning for Language Model Continual Learning](https://mlsanity.com/paper/2310.14152/) (2023)
- [PiSSA: Principal Singular Values and Singular Vectors Adaptation of Large Language Models](https://mlsanity.com/paper/2404.02948/) (2024)
- [DyLoRA: Parameter Efficient Tuning of Pre-trained Models using Dynamic Search-Free Low-Rank Adaptation](https://mlsanity.com/paper/2210.07558/) (2022)
- [Mixture of LoRA Experts](https://mlsanity.com/paper/2404.13628/) (2024)
- [MixLoRA: Enhancing Large Language Models Fine-Tuning with LoRA-based Mixture of Experts](https://mlsanity.com/paper/2404.15159/) (2024)
- [LoRA-GA: Low-Rank Adaptation with Gradient Approximation](https://mlsanity.com/paper/2407.05000/) (2024)
- [LoRA-Pro: Are Low-Rank Adapters Properly Optimized?](https://mlsanity.com/paper/2407.18242/) (2024)
- [Delta-LoRA: Fine-Tuning High-Rank Parameters with the Delta of Low-Rank Matrices](https://mlsanity.com/paper/2309.02411/) (2023)
- [SocialIQA: Commonsense Reasoning about Social Interactions](https://mlsanity.com/paper/1904.09728/) (2019)
- [MELoRA: Mini-Ensemble Low-Rank Adapters for Parameter-Efficient Fine-Tuning](https://mlsanity.com/paper/2402.17263/) (2024)
- [NeuReasoner: Towards Explainable, Controllable, and Unified Reasoning via Mixture-of-Neurons](https://mlsanity.com/paper/2604.02972/) (2026)
- [MSPLoRA: A Multi-Scale Pyramid Low-Rank Adaptation for Efficient Model Fine-Tuning](https://mlsanity.com/paper/2503.21838/) (2025)

## Abstract (arXiv)

Low-Rank Adaptation (LoRA) is a widely used parameter-efficient fine-tuning (PEFT) method for large language models. Under a fixed rank budget, LoRA parameterizes each adapted weight through a single low-dimensional input-side pathway, which may couple heterogeneous behaviors through shared input directions and induce interference during optimization. We propose Static Orthogonal Subspace LoRA (SOS-LoRA), a drop-in extension that reparameterizes a rank-rtot update as a sum of K static (always-on, non-routed) low-rank experts. SOS-LoRA (i) decomposes the total rank across experts, (ii) applies a fixed multi-scale scaling scheme to encourage scale-separated optimization dynamics, and (iii) promotes diverse input-side directions via cross-expert orthogonal initialization and a lightweight regularizer. SOS-LoRA remains fully mergeable, adding no inference-time parameters or latency after merging. Experiments on reasoning and knowledge-intensive benchmarks (Llama 2/3), encoder-based NLU (GLUE), and math reasoning (GSM8K/MATH) show consistent gains over matched-budget LoRA baselines and recent variants. Code is available at https://github.com/llm172/sos-lora.