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Mechanisms of Projective Composition of Diffusion Models
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Gives a formal definition of projective composition for diffusion models and proves when linear score combination (composition operator) with a background distribution yields it, via factorized conditionals and feature-space extensions.
Problem
Linear score combination for composing diffusion models often works empirically (including length-generalization), but there is no formal definition of what correct composition means and no theory of when it succeeds or fails.
Approach
Defines projective composition (distribution whose projected marginals match each component's projected marginals), shows the composition operator with a background distribution produces it under factorized-conditional independence structure, extends to feature spaces via reparameterization equivariance, and proposes a CLIP-embedding orthogonality heuristic to predict composition success.
Key results
- Single-object empty-background composition places the correct number of objects in 98-100% of samples for N=1 to 5 (N=6: 97%).
- Single-object Bayes composition with unconditional background fails: for N=4, 77 of 100 samples contain zero objects in correct locations.
- Bayes-cluttered composition on 1-5 object distributions yields correct counts for N up to 4 at 100%, and 98% for N=5 and N=6.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use an empty rather than unconditional background when composing single-object distributions | Score composition on CLEVR single-object distributions · CLEVR, N=2 through N=6 | images with the expected number of correct objects: N=2: 100 vs 67; N=4: 98 vs 0 | Helped |
| Use an unconditional background with cluttered 1-5 object distributions instead of single-object distributions | Bayes-cluttered score composition · CLEVR, N=2 through N=6 | images with the expected number of correct objects: N=2: 100; N=3: 100; N=4: 100; N=5: 98; N=6: 98 | Helped |
| Compose single-object distributions with an unconditional background rather than an empty background | Score composition on CLEVR single-object distributions · CLEVR, N=4 and N=5 | images with 0 correct objects: N=4: 77; N=5: 66 | Hurt |
Surprising findings
- Bayes composition with an unconditional background often produces no objects at all when composing many concepts.
- Composing non-orthogonal colors fails under diffusion sampling even though theory predicts all color pairs should compose equally well via linear transforms.
Limitations
- Theory assumes factorized conditional independence that holds only approximately in CLEVR and pixel space.
- Orthogonality heuristic is necessary but not sufficient for projective composition.
- No sampling guarantee for non-orthogonal feature-space transforms; composition can be hard to sample.
Experimental setup
Datasets: CLEVR (custom generated) personal dog photos (30 images)
Benchmarks: CLEVR location-conditioned composition (manual counts)
Metrics: manual object-count histogram in correct locations cosine similarity of CLIP mean-difference vectors
Baselines: Bayes composition simple product
Builds on: Bayes composition classifier-free guidance EDM2 CLEVR
Closest papers by meaning
- CoInD: Enabling Logical Compositions in Diffusion Models (ICLR 2025 · similarity 0.88)
- Improving Compositional Generation with Diffusion Models Using Lift Scores (ICML 2025 · similarity 0.87)
- Composition and Alignment of Diffusion Models using Constrained Learning (NeurIPS 2025 · similarity 0.87)
- How Diffusion Models Learn to Factorize and Compose (NeurIPS 2024 · similarity 0.87)
- The Emergence of Reproducibility and Consistency in Diffusion Models (ICML 2024 · similarity 0.87)
- Role Bias in Diffusion Models: Diagnosing and Mitigating through Intermediate Decomposition (NeurIPS 2025 · similarity 0.86)
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