# Subspace Control: Turning Constrained Model Steering into Controllable Spectral Optimization

arXiv: https://arxiv.org/abs/2604.04231 · HTML: https://mlsanity.com/paper/2604.04231/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Machine unlearning, one of 12 directions in LLM safety and security.
- Builds on 51 of its 61 references that are in the map.
- First posted on arXiv 5 Apr 2026.

## Placed in

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

## Builds on (most cited first, 30 of 51)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [Instruction-Following Evaluation for Large Language Models](https://mlsanity.com/paper/2311.07911/) (2023)
- [Editing Models with Task Arithmetic](https://mlsanity.com/paper/2212.04089/) (2022)
- [Parameter-Efficient Fine-Tuning for Large Models: A Comprehensive Survey](https://mlsanity.com/paper/2403.14608/) (2024)
- [The WMDP Benchmark: Measuring and Reducing Malicious Use With Unlearning](https://mlsanity.com/paper/2403.03218/) (2024)
- [Negative Preference Optimization: From Catastrophic Collapse to Effective Unlearning](https://mlsanity.com/paper/2404.05868/) (2024)
- [e-SNLI: Natural Language Inference with Natural Language Explanations](https://mlsanity.com/paper/1812.01193/) (2018)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Explain Yourself! Leveraging Language Models for Commonsense Reasoning](https://mlsanity.com/paper/1906.02361/) (2019)
- [RAGTruth: A Hallucination Corpus for Developing Trustworthy Retrieval-Augmented Language Models](https://mlsanity.com/paper/2401.00396/) (2023)
- [PKU-SafeRLHF: Towards Multi-Level Safety Alignment for LLMs with Human Preference](https://mlsanity.com/paper/2406.15513/) (2024)
- [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)
- [Parameter-Efficient Fine-Tuning Methods for Pretrained Language Models: A Critical Review and Assessment](https://mlsanity.com/paper/2312.12148/) (2023)
- [GLM-4-Voice: Towards Intelligent and Human-Like End-to-End Spoken Chatbot](https://mlsanity.com/paper/2412.02612/) (2024)
- [VoiceBench: Benchmarking LLM-Based Voice Assistants](https://mlsanity.com/paper/2410.17196/) (2024)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [Mitigating the Alignment Tax of RLHF](https://mlsanity.com/paper/2309.06256/) (2023)
- [Task Singular Vectors: Reducing Task Interference in Model Merging](https://mlsanity.com/paper/2412.00081/) (2024)
- [Spirit LM: Interleaved Spoken and Written Language Model](https://mlsanity.com/paper/2402.05755/) (2024)
- [Safety Tax: Safety Alignment Makes Your Large Reasoning Models Less Reasonable](https://mlsanity.com/paper/2503.00555/) (2025)
- [No Task Left Behind: Isotropic Model Merging with Common and Task-Specific Subspaces](https://mlsanity.com/paper/2502.04959/) (2025)
- [Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization](https://mlsanity.com/paper/2503.12645/) (2025)

## Built on by (most cited first, 1 of 1)

- [TOUR: A Trajectory-Level Unlearning Benchmark for Offline Reinforcement Learning](https://mlsanity.com/paper/2607.21111/) (2026)

## Abstract (arXiv)

Foundation models, such as large language models (LLMs), are powerful but often require customization before deployment to satisfy practical constraints such as safety, privacy, and task-specific requirements, leading to "constrained" optimization problems for model steering and adaptation. However, solving such problems remains largely underexplored and is particularly challenging due to interference between the primary objective and constraint objectives during optimization. In this paper, we propose a subspace control framework for constrained model training. Specifically, (i) we first analyze, from a model merging perspective, how spectral cross-task interference arises and show that it can be resolved via a one-shot solution that orthogonalizes the merged subspace; (ii) we establish a connection between this solution and gradient orthogonalization in the spectral optimizer Muon; and (iii) building on these insights, we introduce SIFT (spectral interference-free training), which leverages a localization scheme to selectively intervene during optimization, enabling controllable updates that mitigate objective-constraint conflicts. We evaluate SIFT across four representative applications: (a) machine unlearning, (b) safety alignment, (c) text-to-speech adaptation, and (d) hallucination mitigation. Compared to both control-based and control-free baselines, SIFT consistently achieves substantial and robust performance improvements across all tasks. Code is available at https://github.com/OPTML-Group/SIFT.