# PiCa: Parameter-Efficient Fine-Tuning with Column Space Projection

arXiv: https://arxiv.org/abs/2505.20211 · HTML: https://mlsanity.com/paper/2505.20211/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 40 references that are in the map.
- First posted on arXiv 26 May 2025.

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

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [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)
- [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)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [DreamBooth: Fine Tuning Text-to-Image Diffusion Models for Subject-Driven Generation](https://mlsanity.com/paper/2208.12242/) (2022)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [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)
- [DeBERTaV3: Improving DeBERTa using ELECTRA-Style Pre-Training with Gradient-Disentangled Embedding Sharing](https://mlsanity.com/paper/2111.09543/) (2021)
- [Gemma: Open Models Based on Gemini Research and Technology](https://mlsanity.com/paper/2403.08295/) (2024)
- [DoRA: Weight-Decomposed Low-Rank Adaptation](https://mlsanity.com/paper/2402.09353/) (2024)
- [A Large-scale Study of Representation Learning with the Visual Task Adaptation Benchmark](https://mlsanity.com/paper/1910.04867/) (2019)
- [SVDiff: Compact Parameter Space for Diffusion Fine-Tuning](https://mlsanity.com/paper/2303.11305/) (2023)
- [VeRA: Vector-based Random Matrix Adaptation](https://mlsanity.com/paper/2310.11454/) (2023)
- [Using the Output Embedding to Improve Language Models](https://mlsanity.com/paper/1608.05859/) (2016)
- [PiSSA: Principal Singular Values and Singular Vectors Adaptation of Large Language Models](https://mlsanity.com/paper/2404.02948/) (2024)
- [Controlling Text-to-Image Diffusion by Orthogonal Finetuning](https://mlsanity.com/paper/2306.07280/) (2023)
- [Parameter-Efficient Orthogonal Finetuning via Butterfly Factorization](https://mlsanity.com/paper/2311.06243/) (2023)
- [LoRA vs Full Fine-tuning: An Illusion of Equivalence](https://mlsanity.com/paper/2410.21228/) (2024)
- [Tying Word Vectors and Word Classifiers: A Loss Framework for Language Modeling](https://mlsanity.com/paper/1611.01462/) (2016)
- [Punica: Multi-Tenant LoRA Serving](https://mlsanity.com/paper/2310.18547/) (2023)
- [Tied-Lora: Enhancing parameter efficiency of LoRA with weight tying](https://mlsanity.com/paper/2311.09578/) (2023)
- [SVFT: Parameter-Efficient Fine-Tuning with Singular Vectors](https://mlsanity.com/paper/2405.19597/) (2024)
- [ShareLoRA: Parameter Efficient and Robust Large Language Model Fine-tuning via Shared Low-Rank Adaptation](https://mlsanity.com/paper/2406.10785/) (2024)
- [DiTASK: Multi-Task Fine-Tuning with Diffeomorphic Transformations](https://mlsanity.com/paper/2502.06029/) (2025)
- [Hollowed Net for On-Device Personalization of Text-to-Image Diffusion Models](https://mlsanity.com/paper/2411.01179/) (2024)

## Built on by (most cited first, 6 of 6)

- [Muonp: Muon with Fractional Spectral Powers](https://mlsanity.com/paper/2606.13867/) (2026)
- [Non-vacuous Generalization Bounds for Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2607.14506/) (2026)
- [One Algorithm, Two Goals: Dual Scoring for Parameter and Data Selection in LLM Fine-Tuning](https://mlsanity.com/paper/2605.06166/) (2026)
- [MoNe: Modular Neural Memory for Efficient Long Context Inference](https://mlsanity.com/paper/2608.17616/) (2026)
- [DeCo: Efficient Decouple-to-Couple Learning for Multi-Task Visual Grounding](https://mlsanity.com/paper/2609.24409/) (2026)
- [Frozen Factor or Spectral Band? Disentangling Two Choices in Low-Rank LoRA](https://mlsanity.com/paper/2610.06621/) (2026)

## Abstract (arXiv)

Fine-tuning large foundation models is essential for building expert models tailored to specialized tasks and domains, but fully updating billions of parameters is computationally prohibitive. Reducing the number of trainable parameters using Parameter-Efficient Fine-Tuning (PEFT), such as Low-Rank Adaptation (LoRA), is therefore crucial not only to reduce training costs but also to mitigate storage, caching, and serving overheads during deployment. Prior works, such as Singular Vectors-guided Fine-Tuning (SVFT), have shown that exploiting the geometry of pre-trained weights based on Singular Value Decomposition (SVD) can significantly improve parameter-efficiency, but they lack a solid theoretical foundation. In this paper, we introduce Parameter-Efficient Fine-Tuning with Column Space Projection (PiCa), a novel theoretically grounded PEFT method. We prove that projecting gradients onto the principal column space of pre-trained weights provides an effective inductive bias for adaptation and further enhance parameter efficiency through a novel weight-sharing strategy. Across diverse NLP and vision tasks, PiCa consistently outperforms state-of-the-art baselines under comparable or smaller parameter budgets, demonstrating both theoretical rigor and practical effectiveness.