# Compositional Abilities Emerge Multiplicatively: Exploring Diffusion Models on a Synthetic Task

arXiv: https://arxiv.org/abs/2310.09336 · HTML: https://mlsanity.com/paper/2310.09336/

## Key facts

- Cited by 93 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 95 of its 123 references that are in the map.
- First posted on arXiv 13 Oct 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 95)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [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)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [Finetuned Language Models Are Zero-Shot Learners](https://mlsanity.com/paper/2109.01652/) (2021)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [DreamFusion: Text-to-3D using 2D Diffusion](https://mlsanity.com/paper/2209.14988/) (2022)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [InstructPix2Pix: Learning to Follow Image Editing Instructions](https://mlsanity.com/paper/2211.09800/) (2022)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Do As I Can, Not As I Say: Grounding Language in Robotic Affordances](https://mlsanity.com/paper/2204.01691/) (2022)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Align your Latents: High-Resolution Video Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2304.08818/) (2023)
- [CLEVR: A Diagnostic Dataset for Compositional Language and Elementary Visual Reasoning](https://mlsanity.com/paper/1612.06890/) (2016)

## Built on by (most cited first, 30 of 93)

- [An analytic theory of creativity in convolutional diffusion models](https://mlsanity.com/paper/2412.20292/) (2024)
- [A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data](https://mlsanity.com/paper/2402.16991/) (2024)
- [From Flat to Hierarchical: Extracting Sparse Representations with Matching Pursuit](https://mlsanity.com/paper/2506.03093/) (2025)
- [Generative Trajectory Stitching through Diffusion Composition](https://mlsanity.com/paper/2503.05153/) (2025)
- [Emergence of Hidden Capabilities: Exploring Learning Dynamics in Concept Space](https://mlsanity.com/paper/2406.19370/) (2024)
- [Roll the dice & look before you leap: Going beyond the creative limits of next-token prediction](https://mlsanity.com/paper/2504.15266/) (2025)
- [A Percolation Model of Emergence: Analyzing Transformers Trained on a Formal Language](https://mlsanity.com/paper/2408.12578/) (2024)
- [Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior](https://mlsanity.com/paper/2605.05115/) (2026)
- [How Compositional Generalization and Creativity Improve as Diffusion Models are Trained](https://mlsanity.com/paper/2502.12089/) (2025)
- [ECLIPSE: A Resource-Efficient Text-to-Image Prior for Image Generations](https://mlsanity.com/paper/2312.04655/) (2023)
- [Compositional Capabilities of Autoregressive Transformers: A Study on Synthetic, Interpretable Tasks](https://mlsanity.com/paper/2311.12997/) (2023)
- [When does compositional structure yield compositional generalization? A kernel theory](https://mlsanity.com/paper/2405.16391/) (2024)
- [Representation Shattering in Transformers: A Synthetic Study with Knowledge Editing](https://mlsanity.com/paper/2410.17194/) (2024)
- [Initialization is Critical to Whether Transformers Fit Composite Functions by Reasoning or Memorizing](https://mlsanity.com/paper/2405.05409/) (2024)
- [Can Models Learn Skill Composition from Examples?](https://mlsanity.com/paper/2409.19808/) (2024)
- [Who's in and who's out? A case study of multimodal CLIP-filtering in DataComp](https://mlsanity.com/paper/2405.08209/) (2024)
- [A Survey on Compositional Learning of AI Models: Theoretical and Experimental Practices](https://mlsanity.com/paper/2406.08787/) (2024)
- [Understanding Visual Feature Reliance through the Lens of Complexity](https://mlsanity.com/paper/2407.06076/) (2024)
- [Neural Scaling Laws Rooted in the Data Distribution](https://mlsanity.com/paper/2412.07942/) (2024)
- [Mechanisms of Projective Composition of Diffusion Models](https://mlsanity.com/paper/2502.04549/) (2025)
- [Generative Modeling of Weights: Generalization or Memorization?](https://mlsanity.com/paper/2506.07998/) (2025)
- [How Diffusion Models Learn to Factorize and Compose](https://mlsanity.com/paper/2408.13256/) (2024)
- [An Analysis for Reasoning Bias of Language Models with Small Initialization](https://mlsanity.com/paper/2502.04375/) (2025)
- [Making Reconstruction FID Predictive of Diffusion Generation FID](https://mlsanity.com/paper/2603.05630/) (2026)
- [MetaSlot: Break Through the Fixed Number of Slots in Object-Centric Learning](https://mlsanity.com/paper/2505.20772/) (2025)
- [Decomposing Elements of Problem Solving: What "Math" Does RL Teach?](https://mlsanity.com/paper/2505.22756/) (2025)
- [Relations, Negations, and Numbers: Looking for Logic in Generative Text-to-Image Models](https://mlsanity.com/paper/2411.17066/) (2024)
- [When and How Does CLIP Enable Domain and Compositional Generalization?](https://mlsanity.com/paper/2502.09507/) (2025)
- [Position: Solve Layerwise Linear Models First to Understand Neural Dynamical Phenomena (Neural Collapse, Emergence, Lazy/Rich Regime, and Grokking)](https://mlsanity.com/paper/2502.21009/) (2025)
- [Skews in the Phenomenon Space Hinder Generalization in Text-to-Image Generation](https://mlsanity.com/paper/2403.16394/) (2024)

## Abstract (arXiv)

Modern generative models exhibit unprecedented capabilities to generate extremely realistic data. However, given the inherent compositionality of the real world, reliable use of these models in practical applications requires that they exhibit the capability to compose a novel set of concepts to generate outputs not seen in the training data set. Prior work demonstrates that recent diffusion models do exhibit intriguing compositional generalization abilities, but also fail unpredictably. Motivated by this, we perform a controlled study for understanding compositional generalization in conditional diffusion models in a synthetic setting, varying different attributes of the training data and measuring the model's ability to generate samples out-of-distribution. Our results show: (i) the order in which the ability to generate samples from a concept and compose them emerges is governed by the structure of the underlying data-generating process; (ii) performance on compositional tasks exhibits a sudden "emergence" due to multiplicative reliance on the performance of constituent tasks, partially explaining emergent phenomena seen in generative models; and (iii) composing concepts with lower frequency in the training data to generate out-of-distribution samples requires considerably more optimization steps compared to generating in-distribution samples. Overall, our study lays a foundation for understanding capabilities and compositionality in generative models from a data-centric perspective.