# LIFT the Veil for the Truth: Principal Weights Emerge after Rank Reduction for Reasoning-Focused Supervised Fine-Tuning

arXiv: https://arxiv.org/abs/2506.00772 · HTML: https://mlsanity.com/paper/2506.00772/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 26 of its 59 references that are in the map.
- First posted on arXiv 1 Jun 2025.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 26 of 26)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [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)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training](https://mlsanity.com/paper/2501.17161/) (2025)
- [Dissecting Recall of Factual Associations in Auto-Regressive Language Models](https://mlsanity.com/paper/2304.14767/) (2023)
- [LIMO: Less is More for Reasoning](https://mlsanity.com/paper/2502.03387/) (2025)
- [Movement Pruning: Adaptive Sparsity by Fine-Tuning](https://mlsanity.com/paper/2005.07683/) (2020)
- [Parameter-Efficient Transfer Learning with Diff Pruning](https://mlsanity.com/paper/2012.07463/) (2020)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Assessing the Brittleness of Safety Alignment via Pruning and Low-Rank Modifications](https://mlsanity.com/paper/2402.05162/) (2024)
- [Outlier Weighed Layerwise Sparsity (OWL): A Missing Secret Sauce for Pruning LLMs to High Sparsity](https://mlsanity.com/paper/2310.05175/) (2023)
- [Raise a Child in Large Language Model: Towards Effective and Generalizable Fine-tuning](https://mlsanity.com/paper/2109.05687/) (2021)
- [MiLoRA: Harnessing Minor Singular Components for Parameter-Efficient LLM Finetuning](https://mlsanity.com/paper/2406.09044/) (2024)
- [Stable Rank Normalization for Improved Generalization in Neural Networks and GANs](https://mlsanity.com/paper/1906.04659/) (2019)
- [From Low Rank Gradient Subspace Stabilization to Low-Rank Weights: Observations, Theories, and Applications](https://mlsanity.com/paper/2407.11239/) (2024)
- [A Random Matrix Theory Perspective on the Spectrum of Learned Features and Asymptotic Generalization Capabilities](https://mlsanity.com/paper/2410.18938/) (2024)
- [Outlier-weighed Layerwise Sampling for LLM Fine-tuning](https://mlsanity.com/paper/2405.18380/) (2024)

## Built on by (most cited first, 12 of 12)

- [The Path Not Taken: RLVR Provably Learns Off the Principals](https://mlsanity.com/paper/2511.08567/) (2025)
- [On the Geometry of On-Policy Distillation](https://mlsanity.com/paper/2606.07082/) (2026)
- [HTMuon: Improving Muon via Heavy-Tailed Spectral Correction](https://mlsanity.com/paper/2603.10067/) (2026)
- [K2-Think: A Parameter-Efficient Reasoning System](https://mlsanity.com/paper/2509.07604/) (2025)
- [Eigenspectrum Analysis of Neural Networks without Aspect Ratio Bias](https://mlsanity.com/paper/2506.06280/) (2025)
- [GeoRA: Geometry-Aware Low-Rank Adaptation for RLVR](https://mlsanity.com/paper/2601.09361/) (2026)
- [TeRA: Vector-based Random Tensor Network for High-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2509.03234/) (2025)
- [Constructive Circuit Amplification: Improving Math Reasoning in LLMs via Targeted Sub-Network Updates](https://mlsanity.com/paper/2512.16914/) (2025)
- [RecycleLoRA: Rank-Revealing QR-Based Dual-LoRA Subspace Adaptation for Domain Generalized Semantic Segmentation](https://mlsanity.com/paper/2603.28142/) (2026)
- [One Algorithm, Two Goals: Dual Scoring for Parameter and Data Selection in LLM Fine-Tuning](https://mlsanity.com/paper/2605.06166/) (2026)
- [FuRA: Full-Rank Parameter-Efficient Fine-Tuning with Spectral Preconditioning](https://mlsanity.com/paper/2605.22869/) (2026)
- [Geometric Calibration of Supervised Fine-Tuning](https://mlsanity.com/paper/2610.18634/) (2026)

## Abstract (arXiv)

Recent studies have shown that supervised fine-tuning of LLMs on a small number of high-quality datasets can yield strong reasoning capabilities. However, full fine-tuning (Full FT), while powerful, is computationally expensive and susceptible to overfitting and catastrophic forgetting, particularly when data is limited. Sparse fine-tuning, which previously achieved notable success by updating only a small subset of model parameters, offers a promising trade-off between efficiency and effectiveness. Yet, it has lagged behind in the LLM era due to the difficulty of identifying parameters truly critical for reasoning. In this work, we state that weights with the largest magnitude after low-rank approximation are critical weights for fine-tuning, which we call Principal Weights. Surprisingly, while magnitude-based sparse fine-tuning performs poorly as a baseline on LLM fine-tuning, it becomes highly effective after rank reduction. These insights motivate our method: Low-rank Informed Sparse Fine-Tuning (LIFT). LIFT only updates the top 5% Principal Weights throughout training and consistently achieves better performance on reasoning tasks than Full FT, while maintaining memory efficiency on par with popular parameter-efficient fine-tuning methods. In addition to strong performance on target domains such as arithmetic reasoning, LIFT also retains up to 20% more source-domain knowledge, compared to Full FT and LoRA. Our code is available at: https://github.com/zihanghliu/LIFT.