# Null-Space Constrained Low-Rank Adaptation for Response-Specified Large Language Model Unlearning

arXiv: https://arxiv.org/abs/2606.10989 · HTML: https://mlsanity.com/paper/2606.10989/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Machine unlearning, one of 12 directions in LLM safety and security.
- Builds on 44 of its 51 references that are in the map.
- First posted on arXiv 9 Jun 2026.

## Placed in

- LLM safety and security / [Machine unlearning](https://mlsanity.com/safety/machine-unlearning/)

## Builds on (most cited first, 30 of 44)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [HarmBench: A Standardized Evaluation Framework for Automated Red Teaming and Robust Refusal](https://mlsanity.com/paper/2402.04249/) (2024)
- [Machine Unlearning](https://mlsanity.com/paper/1912.03817/) (2019)
- [Efficient Lifelong Learning with A-GEM](https://mlsanity.com/paper/1812.00420/) (2018)
- [Mass-Editing Memory in a Transformer](https://mlsanity.com/paper/2210.07229/) (2022)
- [Knowledge Neurons in Pretrained Transformers](https://mlsanity.com/paper/2104.08696/) (2021)
- [JailbreakBench: An Open Robustness Benchmark for Jailbreaking Large Language Models](https://mlsanity.com/paper/2404.01318/) (2024)
- [TOFU: A Task of Fictitious Unlearning for LLMs](https://mlsanity.com/paper/2401.06121/) (2024)
- [The WMDP Benchmark: Measuring and Reducing Malicious Use With Unlearning](https://mlsanity.com/paper/2403.03218/) (2024)
- [Zephyr: Direct Distillation of LM Alignment](https://mlsanity.com/paper/2310.16944/) (2023)
- [Negative Preference Optimization: From Catastrophic Collapse to Effective Unlearning](https://mlsanity.com/paper/2404.05868/) (2024)
- [Fast Model Editing at Scale](https://mlsanity.com/paper/2110.11309/) (2021)
- ["Do Anything Now": Characterizing and Evaluating In-The-Wild Jailbreak Prompts on Large Language Models](https://mlsanity.com/paper/2308.03825/) (2023)
- [Who's Harry Potter? Approximate Unlearning in LLMs](https://mlsanity.com/paper/2310.02238/) (2023)
- [LoRA Learns Less and Forgets Less](https://mlsanity.com/paper/2405.09673/) (2024)
- [Large Language Model Unlearning](https://mlsanity.com/paper/2310.10683/) (2023)
- [MUSE: Machine Unlearning Six-Way Evaluation for Language Models](https://mlsanity.com/paper/2407.06460/) (2024)
- [Rethinking Machine Unlearning for Large Language Models](https://mlsanity.com/paper/2402.08787/) (2024)
- [Multilingual Jailbreak Challenges in Large Language Models](https://mlsanity.com/paper/2310.06474/) (2023)
- [AlphaEdit: Null-Space Constrained Knowledge Editing for Language Models](https://mlsanity.com/paper/2410.02355/) (2024)
- [Continual Learning of Context-dependent Processing in Neural Networks](https://mlsanity.com/paper/1810.01256/) (2018)
- [Simplicity Prevails: Rethinking Negative Preference Optimization for LLM Unlearning](https://mlsanity.com/paper/2410.07163/) (2024)
- [Training Networks in Null Space of Feature Covariance for Continual Learning](https://mlsanity.com/paper/2103.07113/) (2021)
- [Machine Unlearning of Pre-trained Large Language Models](https://mlsanity.com/paper/2402.15159/) (2024)
- [Don't Listen To Me: Understanding and Exploring Jailbreak Prompts of Large Language Models](https://mlsanity.com/paper/2403.17336/) (2024)
- [OpenUnlearning: Accelerating LLM Unlearning via Unified Benchmarking of Methods and Metrics](https://mlsanity.com/paper/2506.12618/) (2025)
- [Position: LLM Unlearning Benchmarks are Weak Measures of Progress](https://mlsanity.com/paper/2410.02879/) (2024)

## Abstract (arXiv)

Large language model unlearning aims to suppress designated undesirable knowledge while preserving benign capabilities. Many unlearning objectives focus on suppressing undesired answers, while recent target-guided variants specify replacement behavior but still leave update locality largely unconstrained. This paper introduces Null-Space Constrained Response-Specified Unlearning (NSRU), a projection-constrained low-rank framework for controlled LLM unlearning. NSRU uses an explicitly structured safe target response to specify the desired behavior for each forget query, while suppressing the original undesired content. To localize adaptation, NSRU estimates per-module retain subspaces from benign hidden representations and uses an orthogonal-projected low-rank parameterization to confine LoRA updates to the null space of the retain subspace. The resulting objective jointly optimizes safe-target learning, undesired-response suppression, and retention preservation under this constrained parameterization. We provide a local first-order analysis showing that the projected update reduces retain-side perturbations while preserving editable directions for shaping forget-query behavior. Experiments on TOFU show that NSRU effectively suppresses extractable forget-set knowledge while improving retain QA performance, model utility, and safe-target alignment over representative baselines. On WMDP, NSRU keeps hazardous-domain accuracy near the random-choice region while preserving broad and domain-adjacent MMLU utility. Ablation studies support the complementary roles of safe-target supervision, undesired-response suppression, retention loss, and null-space projected updates, while sensitivity and robustness analyses indicate stable behavior across the tested hyperparameter and prompt variations.