# PiSSA: Principal Singular Values and Singular Vectors Adaptation of Large Language Models

arXiv: https://arxiv.org/abs/2404.02948 · HTML: https://mlsanity.com/paper/2404.02948/

## Key facts

- Cited by 274 papers in the map (data through 14 Oct 2026).
- Builds on 61 of its 73 references that are in the map.
- First posted on arXiv 3 Apr 2024.

## Builds on (most cited first, 30 of 61)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [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)
- [Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena](https://mlsanity.com/paper/2306.05685/) (2023)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [The Power of Scale for Parameter-Efficient Prompt Tuning](https://mlsanity.com/paper/2104.08691/) (2021)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [Program Synthesis with Large Language Models](https://mlsanity.com/paper/2108.07732/) (2021)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [DeBERTaV3: Improving DeBERTa using ELECTRA-Style Pre-Training with Gradient-Disentangled Embedding Sharing](https://mlsanity.com/paper/2111.09543/) (2021)
- [BitFit: Simple Parameter-efficient Fine-tuning for Transformer-based Masked Language-models](https://mlsanity.com/paper/2106.10199/) (2021)
- [WizardLM: Empowering large pre-trained language models to follow complex instructions](https://mlsanity.com/paper/2304.12244/) (2023)
- [Gemma: Open Models Based on Gemini Research and Technology](https://mlsanity.com/paper/2403.08295/) (2024)
- [Towards a Unified View of Parameter-Efficient Transfer Learning](https://mlsanity.com/paper/2110.04366/) (2021)
- [DeepSeekMoE: Towards Ultimate Expert Specialization in Mixture-of-Experts Language Models](https://mlsanity.com/paper/2401.06066/) (2024)
- [AdapterFusion: Non-Destructive Task Composition for Transfer Learning](https://mlsanity.com/paper/2005.00247/) (2020)
- [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)
- [WizardCoder: Empowering Code Large Language Models with Evol-Instruct](https://mlsanity.com/paper/2306.08568/) (2023)

## Built on by (most cited first, 30 of 274)

- [LoRA Learns Less and Forgets Less](https://mlsanity.com/paper/2405.09673/) (2024)
- [SVDQuant: Absorbing Outliers by Low-Rank Components for 4-Bit Diffusion Models](https://mlsanity.com/paper/2411.05007/) (2024)
- [LoRA vs Full Fine-tuning: An Illusion of Equivalence](https://mlsanity.com/paper/2410.21228/) (2024)
- [Cartridges: Lightweight and general-purpose long context representations via self-study](https://mlsanity.com/paper/2506.06266/) (2025)
- [Relaxed Recursive Transformers: Effective Parameter Sharing with Layer-wise LoRA](https://mlsanity.com/paper/2410.20672/) (2024)
- [LoRA-XS: Low-Rank Adaptation with Extremely Small Number of Parameters](https://mlsanity.com/paper/2405.17604/) (2024)
- [LoRA-Pro: Are Low-Rank Adapters Properly Optimized?](https://mlsanity.com/paper/2407.18242/) (2024)
- [A Survey on LoRA of Large Language Models](https://mlsanity.com/paper/2407.11046/) (2024)
- [The Path Not Taken: RLVR Provably Learns Off the Principals](https://mlsanity.com/paper/2511.08567/) (2025)
- [SVFT: Parameter-Efficient Fine-Tuning with Singular Vectors](https://mlsanity.com/paper/2405.19597/) (2024)
- [The Impact of Initialization on LoRA Finetuning Dynamics](https://mlsanity.com/paper/2406.08447/) (2024)
- [MC-LLaVA: Multi-Concept Personalized Vision-Language Model](https://mlsanity.com/paper/2411.11706/) (2024)
- [Parameter-Efficient Fine-Tuning for Foundation Models](https://mlsanity.com/paper/2501.13787/) (2025)
- [CorDA: Context-Oriented Decomposition Adaptation of Large Language Models for Task-Aware Parameter-Efficient Fine-tuning](https://mlsanity.com/paper/2406.05223/) (2024)
- [SaLoRA: Safety-Alignment Preserved Low-Rank Adaptation](https://mlsanity.com/paper/2501.01765/) (2025)
- [LoRA-FAIR: Federated LoRA Fine-Tuning with Aggregation and Initialization Refinement](https://mlsanity.com/paper/2411.14961/) (2024)
- [Towards Robust and Parameter-Efficient Knowledge Unlearning for LLMs](https://mlsanity.com/paper/2408.06621/) (2024)
- [RandLoRA: Full-rank parameter-efficient fine-tuning of large models](https://mlsanity.com/paper/2502.00987/) (2025)
- [How Far Are We From AGI: Are LLMs All We Need?](https://mlsanity.com/paper/2405.10313/) (2024)
- [Apple Intelligence Foundation Language Models: Tech Report 2025](https://mlsanity.com/paper/2507.13575/) (2025)
- [RobustMerge: Parameter-Efficient Model Merging for MLLMs with Direction Robustness](https://mlsanity.com/paper/2502.17159/) (2025)
- [Learning on LoRAs: GL-Equivariant Processing of Low-Rank Weight Spaces for Large Finetuned Models](https://mlsanity.com/paper/2410.04207/) (2024)
- [Compress then Serve: Serving Thousands of LoRA Adapters with Little Overhead](https://mlsanity.com/paper/2407.00066/) (2024)
- [Reassessing Layer Pruning in LLMs: New Insights and Methods](https://mlsanity.com/paper/2411.15558/) (2024)
- [SoMA: Singular Value Decomposed Minor Components Adaptation for Domain Generalizable Representation Learning](https://mlsanity.com/paper/2412.04077/) (2024)
- [QERA: an Analytical Framework for Quantization Error Reconstruction](https://mlsanity.com/paper/2410.06040/) (2024)
- [Initialization using Update Approximation is a Silver Bullet for Extremely Efficient Low-Rank Fine-Tuning](https://mlsanity.com/paper/2411.19557/) (2024)
- [AuroRA: Breaking Low-Rank Bottleneck of LoRA with Nonlinear Mapping](https://mlsanity.com/paper/2505.18738/) (2025)
- [PoLAR: Polar-Decomposed Low-Rank Adapter Representation](https://mlsanity.com/paper/2506.03133/) (2025)
- [LoRI: Reducing Cross-Task Interference in Multi-Task Low-Rank Adaptation](https://mlsanity.com/paper/2504.07448/) (2025)

## Abstract (arXiv)

To parameter-efficiently fine-tune (PEFT) large language models (LLMs), the low-rank adaptation (LoRA) method approximates the model changes Delta W ∈ ℝm × n through the product of two matrices A ∈ ℝm × r and B ∈ ℝr × n, where r ll min(m, n), A is initialized with Gaussian noise, and B with zeros. LoRA freezes the original model W and updates the "Noise & Zero" adapter, which may lead to slow convergence. To overcome this limitation, we introduce Principal Singular values and Singular vectors Adaptation (PiSSA). PiSSA shares the same architecture as LoRA, but initializes the adaptor matrices A and B with the principal components of the original matrix W, and put the remaining components into a residual matrix Wres ∈ ℝm × n which is frozen during fine-tuning. Compared to LoRA, PiSSA updates the principal components while freezing the "residual" parts, allowing faster convergence and enhanced performance. Comparative experiments of PiSSA and LoRA across 12 different models, ranging from 184M to 70B, encompassing 5 NLG and 8 NLU tasks, reveal that PiSSA consistently outperforms LoRA under identical experimental setups. On the GSM8K benchmark, Mistral-7B fine-tuned with PiSSA achieves an accuracy of 72.86%, surpassing LoRA's 67.7% by 5.16%. Due to the same architecture, PiSSA is also compatible with quantization to further reduce the memory requirement of fine-tuning. Compared to QLoRA, QPiSSA exhibits smaller quantization errors in the initial stages. Fine-tuning LLaMA-3-70B on GSM8K, QPiSSA attains an accuracy of 86.05%, exceeding the performances of QLoRA at 81.73%. Leveraging a fast SVD technique, PiSSA can be initialized in only a few seconds, presenting a negligible cost for transitioning from LoRA to PiSSA. Code is available at https://github.com/GraphPKU/PiSSA.