# Closed-Form Concept Erasure via Double Projections

arXiv: https://arxiv.org/abs/2604.10032 · HTML: https://mlsanity.com/paper/2604.10032/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Robustness, privacy and adversarial aspects, one of 14 directions in Diffusion models.
- Builds on 57 of its 72 references that are in the map.
- First posted on arXiv 11 Apr 2026.

## Placed in

- Diffusion models / [Robustness, privacy and adversarial aspects](https://mlsanity.com/diffusion/robustness-privacy-and-adversarial-aspects/)

## Builds on (most cited first, 30 of 57)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [The Unreasonable Effectiveness of Deep Features as a Perceptual Metric](https://mlsanity.com/paper/1801.03924/) (2018)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [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)
- [Extracting Training Data from Large Language Models](https://mlsanity.com/paper/2012.07805/) (2020)
- [Scaling Autoregressive Models for Content-Rich Text-to-Image Generation](https://mlsanity.com/paper/2206.10789/) (2022)
- [Multi-Concept Customization of Text-to-Image Diffusion](https://mlsanity.com/paper/2212.04488/) (2022)
- [FLUX.1 Kontext: Flow Matching for In-Context Image Generation and Editing in Latent Space](https://mlsanity.com/paper/2506.15742/) (2025)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Quantifying Memorization Across Neural Language Models](https://mlsanity.com/paper/2202.07646/) (2022)
- [Extracting Training Data from Diffusion Models](https://mlsanity.com/paper/2301.13188/) (2023)
- [Erasing Concepts from Diffusion Models](https://mlsanity.com/paper/2303.07345/) (2023)
- [Safe Latent Diffusion: Mitigating Inappropriate Degeneration in Diffusion Models](https://mlsanity.com/paper/2211.05105/) (2022)
- [Diffusion Art or Digital Forgery? Investigating Data Replication in Diffusion Models](https://mlsanity.com/paper/2212.03860/) (2022)
- [Unified Concept Editing in Diffusion Models](https://mlsanity.com/paper/2308.14761/) (2023)
- [InstantBooth: Personalized Text-to-Image Generation without Test-Time Finetuning](https://mlsanity.com/paper/2304.03411/) (2023)
- [Ablating Concepts in Text-to-Image Diffusion Models](https://mlsanity.com/paper/2303.13516/) (2023)
- [MACE: Mass Concept Erasure in Diffusion Models](https://mlsanity.com/paper/2403.06135/) (2024)
- [Forget-Me-Not: Learning to Forget in Text-to-Image Diffusion Models](https://mlsanity.com/paper/2303.17591/) (2023)
- [Red-Teaming the Stable Diffusion Safety Filter](https://mlsanity.com/paper/2210.04610/) (2022)
- [Easily Accessible Text-to-Image Generation Amplifies Demographic Stereotypes at Large Scale](https://mlsanity.com/paper/2211.03759/) (2022)

## Built on by (most cited first, 3 of 3)

- [Where Concept Erasure Should Occur: Concept-Layer Alignment in Text-to-Video Diffusion Models](https://mlsanity.com/paper/2605.25941/) (2026)
- [SCOPE: Entanglement Frontier Escape for Source-Free Class Unlearning](https://mlsanity.com/paper/2608.02058/) (2026)
- [Motion Concept Unlearning in Video Diffusion Models](https://mlsanity.com/paper/2609.36832/) (2026)

## Abstract (arXiv)

While modern generative models such as diffusion-based architectures have enabled impressive creative capabilities, they also raise important safety and ethical risks. These concerns have led to growing interest in concept erasure, the process of removing unwanted concepts from model representations. Existing approaches often achieve strong erasure performance but rely on iterative optimization and may inadvertently distort unrelated concepts. In this work, we present a simple yet principled alternative: a linear transformation framework that achieves concept erasure analytically, without any training. Our method adapts a pretrained model through two sequential, closed-form steps: first, computing a proxy projection of the target concept, and second, applying a constrained transformation within the left null space of known concept directions. This design yields a deterministic and geometrically interpretable procedure for safe, efficient, and theory-grounded concept removal. Across a wide range of experiments, including object and style erasure on multiple Stable Diffusion variants and the flow-matching model (FLUX), our approach matches or surpasses the performance of state-of-the-art methods while preserving non-target concepts more faithfully. Requiring only a few seconds to apply, it offers a lightweight and drop-in tool for controlled model editing, advancing the goal of safer and more responsible generative models.