# Parameter-Efficient Orthogonal Finetuning via Butterfly Factorization

arXiv: https://arxiv.org/abs/2311.06243 · HTML: https://mlsanity.com/paper/2311.06243/

## Key facts

- Cited by 114 papers in the map (data through 14 Oct 2026).
- Builds on 73 of its 104 references that are in the map.
- First posted on arXiv 10 Nov 2023.

## Builds on (most cited first, 30 of 73)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [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)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Segment Anything](https://mlsanity.com/paper/2304.02643/) (2023)
- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Adding Conditional Control to Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.05543/) (2023)
- [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)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [DreamBooth: Fine Tuning Text-to-Image Diffusion Models for Subject-Driven Generation](https://mlsanity.com/paper/2208.12242/) (2022)
- [An Image is Worth One Word: Personalizing Text-to-Image Generation using Textual Inversion](https://mlsanity.com/paper/2208.01618/) (2022)
- [Visual Prompt Tuning](https://mlsanity.com/paper/2203.12119/) (2022)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [T2I-Adapter: Learning Adapters to Dig out More Controllable Ability for Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.08453/) (2023)
- [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)
- [Multi-Concept Customization of Text-to-Image Diffusion](https://mlsanity.com/paper/2212.04488/) (2022)
- [Semantic Understanding of Scenes through the ADE20K Dataset](https://mlsanity.com/paper/1608.05442/) (2016)
- [Few-Shot Parameter-Efficient Fine-Tuning is Better and Cheaper than In-Context Learning](https://mlsanity.com/paper/2205.05638/) (2022)
- [Towards a Unified View of Parameter-Efficient Transfer Learning](https://mlsanity.com/paper/2110.04366/) (2021)

## Built on by (most cited first, 30 of 114)

- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [SWIFT:A Scalable lightWeight Infrastructure for Fine-Tuning](https://mlsanity.com/paper/2408.05517/) (2024)
- [ReFT: Representation Finetuning for Language Models](https://mlsanity.com/paper/2404.03592/) (2024)
- [Asymmetry in Low-Rank Adapters of Foundation Models](https://mlsanity.com/paper/2402.16842/) (2024)
- [SVFT: Parameter-Efficient Fine-Tuning with Singular Vectors](https://mlsanity.com/paper/2405.19597/) (2024)
- [Riemannian Preconditioned LoRA for Fine-Tuning Foundation Models](https://mlsanity.com/paper/2402.02347/) (2024)
- [Towards Robust and Parameter-Efficient Knowledge Unlearning for LLMs](https://mlsanity.com/paper/2408.06621/) (2024)
- [Can Large Language Models Understand Symbolic Graphics Programs?](https://mlsanity.com/paper/2408.08313/) (2024)
- [FlyLoRA: Boosting Task Decoupling and Parameter Efficiency via Implicit Rank-Wise Mixture-of-Experts](https://mlsanity.com/paper/2510.08396/) (2025)
- [Efficient Diversity-Preserving Diffusion Alignment via Gradient-Informed GFlowNets](https://mlsanity.com/paper/2412.07775/) (2024)
- [VERA: Explainable Video Anomaly Detection via Verbalized Learning of Vision-Language Models](https://mlsanity.com/paper/2412.01095/) (2024)
- [ChatGarment: Garment Estimation, Generation and Editing via Large Language Models](https://mlsanity.com/paper/2412.17811/) (2024)
- [Spectral Adapter: Fine-Tuning in Spectral Space](https://mlsanity.com/paper/2405.13952/) (2024)
- [Personalized Image Generation with Deep Generative Models: A Decade Survey](https://mlsanity.com/paper/2502.13081/) (2025)
- [PoLAR: Polar-Decomposed Low-Rank Adapter Representation](https://mlsanity.com/paper/2506.03133/) (2025)
- [PA-SAM: Prompt Adapter SAM for High-Quality Image Segmentation](https://mlsanity.com/paper/2401.13051/) (2024)
- [ExPLoRA: Parameter-Efficient Extended Pre-Training to Adapt Vision Transformers under Domain Shifts](https://mlsanity.com/paper/2406.10973/) (2024)
- [Bridging The Gap between Low-rank and Orthogonal Adaptation via Householder Reflection Adaptation](https://mlsanity.com/paper/2405.17484/) (2024)
- [PACE: Marrying generalization in PArameter-efficient fine-tuning with Consistency rEgularization](https://mlsanity.com/paper/2409.17137/) (2024)
- [Value Gradient Guidance for Flow Matching Alignment](https://mlsanity.com/paper/2512.05116/) (2025)
- [NoRA: Nested Low-Rank Adaptation for Efficient Fine-Tuning Large Models](https://mlsanity.com/paper/2408.10280/) (2024)
- [LoRA meets Riemannion: Muon Optimizer for Parametrization-independent Low-Rank Adapters](https://mlsanity.com/paper/2507.12142/) (2025)
- [LoRETTA: Low-Rank Economic Tensor-Train Adaptation for Ultra-Low-Parameter Fine-Tuning of Large Language Models](https://mlsanity.com/paper/2402.11417/) (2024)
- [Generating Symbolic World Models via Test-time Scaling of Large Language Models](https://mlsanity.com/paper/2502.04728/) (2025)
- [Group and Shuffle: Efficient Structured Orthogonal Parametrization](https://mlsanity.com/paper/2406.10019/) (2024)
- [Quantum-PEFT: Ultra parameter-efficient fine-tuning](https://mlsanity.com/paper/2503.05431/) (2025)
- [Sharp Generalization Bounds for Foundation Models with Asymmetric Randomized Low-Rank Adapters](https://mlsanity.com/paper/2506.14530/) (2025)
- [Nabla-R2D3: Effective and Efficient 3D Diffusion Alignment with 2D Rewards](https://mlsanity.com/paper/2506.15684/) (2025)
- [3-in-1: 2D Rotary Adaptation for Efficient Finetuning, Efficient Batching and Composability](https://mlsanity.com/paper/2409.00119/) (2024)
- [Parameter-Efficient Fine-Tuning of Large Language Models using Semantic Knowledge Tuning](https://mlsanity.com/paper/2410.08598/) (2024)

## Abstract (arXiv)

Large foundation models are becoming ubiquitous, but training them from scratch is prohibitively expensive. Thus, efficiently adapting these powerful models to downstream tasks is increasingly important. In this paper, we study a principled finetuning paradigm -- Orthogonal Finetuning (OFT) -- for downstream task adaptation. Despite demonstrating good generalizability, OFT still uses a fairly large number of trainable parameters due to the high dimensionality of orthogonal matrices. To address this, we start by examining OFT from an information transmission perspective, and then identify a few key desiderata that enable better parameter-efficiency. Inspired by how the Cooley-Tukey fast Fourier transform algorithm enables efficient information transmission, we propose an efficient orthogonal parameterization using butterfly structures. We apply this parameterization to OFT, creating a novel parameter-efficient finetuning method, called Orthogonal Butterfly (BOFT). By subsuming OFT as a special case, BOFT introduces a generalized orthogonal finetuning framework. Finally, we conduct an extensive empirical study of adapting large vision transformers, large language models, and text-to-image diffusion models to various downstream tasks in vision and language.