# Concept Algebra for (Score-Based) Text-Controlled Generative Models

arXiv: https://arxiv.org/abs/2302.03693 · HTML: https://mlsanity.com/paper/2302.03693/

## Key facts

- Cited by 59 papers in the map (data through 14 Oct 2026).
- Placed in Image generation, text-to-image and editing, one of 14 directions in Diffusion models.
- Builds on 29 of its 42 references that are in the map.
- First posted on arXiv 7 Feb 2023.

## Placed in

- Diffusion models / [Image generation, text-to-image and editing](https://mlsanity.com/diffusion/image-generation-text-to-image-and-editing/)

## Builds on (most cited first, 29 of 29)

- [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)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Man is to Computer Programmer as Woman is to Homemaker? Debiasing Word Embeddings](https://mlsanity.com/paper/1607.06520/) (2016)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [DiffEdit: Diffusion-based semantic image editing with mask guidance](https://mlsanity.com/paper/2210.11427/) (2022)
- [Compositional Visual Generation with Composable Diffusion Models](https://mlsanity.com/paper/2206.01714/) (2022)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022)
- [Diffusion Models already have a Semantic Latent Space](https://mlsanity.com/paper/2210.10960/) (2022)
- [Self-Supervised Learning with Data Augmentations Provably Isolates Content from Style](https://mlsanity.com/paper/2106.04619/) (2021)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning](https://mlsanity.com/paper/1805.08651/) (2018)
- [Unsupervised Feature Extraction by Time-Contrastive Learning and Nonlinear ICA](https://mlsanity.com/paper/1605.06336/) (2016)
- [Lipstick on a Pig: Debiasing Methods Cover up Systematic Gender Biases in Word Embeddings But do not Remove Them](https://mlsanity.com/paper/1903.03862/) (2019)
- [Contrastive Learning Inverts the Data Generating Process](https://mlsanity.com/paper/2102.08850/) (2021)
- [word2vec Explained: deriving Mikolov et al.'s negative-sampling word-embedding method](https://mlsanity.com/paper/1402.3722/) (2014)
- [Learning to Compose Visual Relations](https://mlsanity.com/paper/2111.09297/) (2021)
- [Unsupervised Learning of Compositional Energy Concepts](https://mlsanity.com/paper/2111.03042/) (2021)
- [Analogies Explained: Towards Understanding Word Embeddings](https://mlsanity.com/paper/1901.09813/) (2019)
- [Learning identifiable and interpretable latent models of high-dimensional neural activity using pi-VAE](https://mlsanity.com/paper/2011.04798/) (2020)
- [DCI-ES: An Extended Disentanglement Framework with Connections to Identifiability](https://mlsanity.com/paper/2210.00364/) (2022)
- [Unite and Conquer: Plug & Play Multi-Modal Synthesis using Diffusion Models](https://mlsanity.com/paper/2212.00793/) (2022)

## Built on by (most cited first, 30 of 59)

- [Programming Refusal with Conditional Activation Steering](https://mlsanity.com/paper/2409.05907/) (2024)
- [The Geometry of Categorical and Hierarchical Concepts in Large Language Models](https://mlsanity.com/paper/2406.01506/) (2024)
- [Analyzing the Generalization and Reliability of Steering Vectors](https://mlsanity.com/paper/2407.12404/) (2024)
- [A Practical Review of Mechanistic Interpretability for Transformer-Based Language Models](https://mlsanity.com/paper/2407.02646/) (2024)
- [From Flat to Hierarchical: Extracting Sparse Representations with Matching Pursuit](https://mlsanity.com/paper/2506.03093/) (2025)
- [Continuous, Subject-Specific Attribute Control in T2I Models by Identifying Semantic Directions](https://mlsanity.com/paper/2403.17064/) (2024)
- [Model Editing as a Robust and Denoised variant of DPO: A Case Study on Toxicity](https://mlsanity.com/paper/2405.13967/) (2024)
- [Emergence of Hidden Capabilities: Exploring Learning Dynamics in Concept Space](https://mlsanity.com/paper/2406.19370/) (2024)
- [Overcoming Sparsity Artifacts in Crosscoders to Interpret Chat-Tuning](https://mlsanity.com/paper/2504.02922/) (2025)
- [Controllable Context Sensitivity and the Knob Behind It](https://mlsanity.com/paper/2411.07404/) (2024)
- [The Geometry of Reasoning: Flowing Logics in Representation Space](https://mlsanity.com/paper/2510.09782/) (2025)
- [PaCE: Parsimonious Concept Engineering for Large Language Models](https://mlsanity.com/paper/2406.04331/) (2024)
- [Uncovering Meanings of Embeddings via Partial Orthogonality](https://mlsanity.com/paper/2310.17611/) (2023)
- [Global Counterfactual Directions](https://mlsanity.com/paper/2404.12488/) (2024)
- [Mechanisms of Projective Composition of Diffusion Models](https://mlsanity.com/paper/2502.04549/) (2025)
- [Compositional Risk Minimization](https://mlsanity.com/paper/2410.06303/) (2024)
- [On the Emergence of Linear Analogies in Word Embeddings](https://mlsanity.com/paper/2505.18651/) (2025)
- [Does Data Scaling Lead to Visual Compositional Generalization?](https://mlsanity.com/paper/2507.07102/) (2025)
- [Diffusion Counterfactual Generation with Semantic Abduction](https://mlsanity.com/paper/2506.07883/) (2025)
- [Fair Generation without Unfair Distortions: Debiasing Text-to-Image Generation with Entanglement-Free Attention](https://mlsanity.com/paper/2506.13298/) (2025)
- [Automated Discovery of Pairwise Interactions from Unstructured Data](https://mlsanity.com/paper/2409.07594/) (2024)
- [Closed-Form Training Dynamics Reveal Learned Features and Linear Structure in Word2Vec-like Models](https://mlsanity.com/paper/2502.09863/) (2025)
- [Responsible Diffusion Models via Constraining Text Embeddings within Safe Regions](https://mlsanity.com/paper/2505.15427/) (2025)
- [Semantic Surgery: Zero-Shot Concept Erasure in Diffusion Models](https://mlsanity.com/paper/2510.22851/) (2025)
- [Towards Compositionality in Concept Learning](https://mlsanity.com/paper/2406.18534/) (2024)
- [Concept-based Analysis of Neural Networks via Vision-Language Models](https://mlsanity.com/paper/2403.19837/) (2024)
- [pOps: Photo-Inspired Diffusion Operators](https://mlsanity.com/paper/2406.01300/) (2024)
- [On Fairness of Unified Multimodal Large Language Model for Image Generation](https://mlsanity.com/paper/2502.03429/) (2025)
- [Vision-and-Language Training Helps Deploy Taxonomic Knowledge but Does Not Fundamentally Alter It](https://mlsanity.com/paper/2507.13328/) (2025)
- [Mitigating Shortcut Learning with Diffusion Counterfactuals and Diverse Ensembles](https://mlsanity.com/paper/2311.16176/) (2023)

## Abstract (arXiv)

This paper concerns the structure of learned representations in text-guided generative models, focusing on score-based models. A key property of such models is that they can compose disparate concepts in a `disentangled' manner. This suggests these models have internal representations that encode concepts in a `disentangled' manner. Here, we focus on the idea that concepts are encoded as subspaces of some representation space. We formalize what this means, show there's a natural choice for the representation, and develop a simple method for identifying the part of the representation corresponding to a given concept. In particular, this allows us to manipulate the concepts expressed by the model through algebraic manipulation of the representation. We demonstrate the idea with examples using Stable Diffusion. Code in https://github.com/zihao12/concept-algebra-code