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Synergy Between Sufficient Changes and Sparse Mixing Procedure for Disentangled Representation Learning
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Shows that sufficient distribution changes across auxiliary-variable domains and a sparse mixing (Jacobian sparsity) constraint complement each other for identifiability of disentangled latents, and implements this as CG-VAE and CG-GAN.
Problem
Identifying latent variables in nonlinear ICA / disentangled representation learning needs strong assumptions: sufficient distribution changes across many domains (about 2n+1) or structural sparsity of the mixing function, and each may be violated in practice.
Approach
Sparse mixing constraints conditioned on auxiliary variables give conditional independences between latents and observations, which constrain the mapping from estimated to true latents and compensate for insufficient domain changes; conversely, sufficient changes help when sparsity is partially violated. A domain-encoding flow plus an l1 penalty on the decoder Jacobian implements this.
Key results
- On synthetic dataset A with 2 domains, CG-VAE reaches MCC 0.953 (0.007), versus 0.869 for iMSDA and 0.756 for iVAE.
- On MNIST two-domain generation, CG-GAN achieves FID 8.74 versus 16.6 for i-StyleGAN and 117.64 for StyleGAN2-ADA.
- On CelebA two-domain generation, CG-GAN reaches FID 2.57, slightly better than i-StyleGAN (2.65) and StyleGAN2-ADA (3.57).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CG-VAE rather than iMSDA | synthetic disentangled representation learning · Dataset A (2 domains) | MCC: 0.869 (0.002) -> 0.953 (0.007) | Helped |
| Use CG-VAE rather than iMSDA | synthetic disentangled representation learning · Dataset A (4 domains) | MCC: 0.881 (0.043) -> 0.952 (0.017) | Helped |
| Use CG-VAE rather than iMSDA | synthetic disentangled representation learning · Dataset A (6 domains) | MCC: 0.936 (0.025) -> 0.956 (0.016) | Helped |
| Use CG-VAE rather than iMSDA | synthetic disentangled representation learning · Dataset A (8 domains) | MCC: 0.959 (0.017) -> 0.971 (0.009) | Helped |
| Remove the sparse mixing constraint L_s (ablation) | CG-VAE · synthetic datasets with fewer auxiliary domains | MCC | Hurt |
| Remove the sparse mixing constraint L_s (ablation) | CG-VAE · synthetic datasets (one domain) | MCC | Hurt |
| Remove the learnable mask restriction (ablation) | CG-GAN · CelebA (2 domains) | FID: 2.57 -> 2.60 | Hurt |
| Remove the learnable mask restriction (ablation) | CG-GAN · MNIST (2 domains) | Joint-FID: 67.04 -> 81.53 | Hurt |
| Use CG-GAN rather than i-StyleGAN | two-domain image generation · CelebA | FID: 2.65 -> 2.57 | Helped |
| Use CG-GAN rather than i-StyleGAN | two-domain image generation · MNIST | FID: 16.6 -> 8.74 | Helped |
| Use CG-GAN rather than i-StyleGAN | two-domain image generation · MNIST | Joint-FID: 107.39 -> 67.04 | Helped |
Surprising findings
- CG-VAE still outperforms its no-sparsity variant (CG-VAE-S) even with only a single domain, indicating sparsity alone yields partial identifiability.
- CG-VAE performs worst among the real-world image generation methods, attributed to the Gaussian prior assumption of VAEs.
Limitations
- Empirical study focuses on visual disentanglement; more complex real-world applications not yet tested.
- Theory requires assumptions such as linear independence of derivative vectors and minimal-edge mixing solutions, and the sparsity penalty relies on forward finite-difference approximation for images.
Experimental setup
Datasets: CelebA MNIST synthetic MLP-generated data (Datasets A, B, C)
Benchmarks: synthetic datasets A/B/C CelebA MNIST
Metrics: MCC completeness disentanglement score informativeness R2 MSE FID DIPD Joint-FID
Baselines: beta-VAE FactorVAE SlowVAE iVAE iMSDA TGAN StyleGAN2-ADA i-StyleGAN
Builds on: iVAE iMSDA i-StyleGAN StyleGAN2-ADA structural sparsity identifiability
Closest papers by meaning
- On Causally Disentangled Representations (AAAI 2022 · similarity 0.90)
- $\alpha$TC-VAE: On the relationship between Disentanglement and Diversity (ICLR 2024 · similarity 0.89)
- Can Diffusion Models Disentangle? A Theoretical Perspective (NeurIPS 2025 · similarity 0.88)
- Multi-View Causal Representation Learning with Partial Observability (ICLR 2024 · similarity 0.88)
- Transferring disentangled representations: bridging the gap between synthetic and real images (NeurIPS 2024 · similarity 0.88)
- Learning Linear Causal Representations from General Environments: Identifiability and Intrinsic Ambiguity (NeurIPS 2024 · similarity 0.88)
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