Can Diffusion Models Disentangle? A Theoretical Perspective (2504.00220v1)

Abstract: This paper presents a novel theoretical framework for understanding how diffusion models can learn disentangled representations. Within this framework, we establish identifiability conditions for general disentangled latent variable models, analyze training dynamics, and derive sample complexity bounds for disentangled latent subspace models. To validate our theory, we conduct disentanglement experiments across diverse tasks and modalities, including subspace recovery in latent subspace Gaussian mixture models, image colorization, image denoising, and voice conversion for speech classification. Additionally, our experiments show that training strategies inspired by our theory, such as style guidance regularization, consistently enhance disentanglement performance.

• The paper establishes theoretical identifiability conditions that enable diffusion models to separate latent content from style using an information-regularized score-matching objective.
• It introduces a novel style-guided score-matching objective and mix-and-match loss to ensure content preservation and effective disentanglement in unpaired data settings.
• Empirical results on synthetic data, image tasks, and speech conversion demonstrate the practical benefits and enhanced performance of the proposed disentanglement framework.

This paper, "Can Diffusion Models Disentangle? A Theoretical Perspective" (2504.00220), investigates whether and how diffusion models (DMs) can learn disentangled representations, focusing on the common content-style disentanglement problem. The work provides theoretical foundations and empirical validation for using DMs in tasks requiring separation of latent factors, particularly when only unpaired data is available (e.g., style labels are known, but content labels are not, or vice-versa).

Key Contributions:

1. Theoretical Identifiability: Establishes conditions under which a DM trained with an information-regularized score-matching objective can provably identify disentangled latent variables (content and style) in a general two-variable setting.
2. Novel Training Objective & Analysis: Proposes and analyzes a "style-guided score-matching objective" tailored for DM-based disentanglement. For Latent Subspace Models (LSMs), it analyzes the training dynamics and provides sample complexity bounds.
3. Empirical Validation: Demonstrates the theory's practical relevance through experiments on synthetic (Gaussian Mixture Models) and real-world data (image colorization/denoising on MNIST/CIFAR-10, voice conversion for speech emotion classification on IEMOCAP).

Core Concepts and Theoretical Framework:

• Problem Setup: Assumes data $X$ is generated from latent content $Z$ and observable style $G$ via $X = \alpha \bar{\psi}(Z, G) + N$, where $N$ is Gaussian noise. Key assumptions include $Z \perp G$ (disentanglement) and $Z-X-G$ forming a Markov chain (conditional disentanglement).
• Disentanglement Metrics: Uses $\epsilon$-disentanglement (low mutual information $I(Z;G) \leq \epsilon$) and $(\epsilon, \phi, p)$-editability (ability to swap styles using function $\phi$ while preserving the overall data distribution $p$ within $\epsilon$ total variation distance).
• DM-based Disentanglement (General Case):
  • Proposes using a conditional DM where the reverse diffusion process conditions on the style $G$ and a learnable bottleneck variable $\mathcal{Z} = z_{\phi}(X_{t_0})$ extracted from an early noisy state $X_{t_0}$ (Figure 1).
  • Introduces a regularized conditional score matching objective $L^{\gamma,\rho}_c$ (Eq. 7) that adds a penalty term $\gamma (I(z_{\phi}(X_{t_0});X)-\rho)_+$ to the standard score matching loss $L_c$. This regularizer controls the mutual information between the bottleneck and the input $X$.
  • Theorem 3.3: Shows that minimizing $L^{\gamma,\rho}_c$ under specific conditions (Lipschitz scores, appropriate hyperparameters $\gamma, \rho$) leads to the bottleneck $\mathcal{Z}$ being $O(\epsilon)$-disentangled from the style $G$.
  • Theorem 3.4: Extends this to show $(O(\sqrt{\epsilon}), \psi_{\theta^*}, p)$-editability using the learned generator $\psi_{\theta^*}$.
• Addressing Content Distortion:
  • Highlights that $I(\mathcal{Z};G) \approx 0$ doesn't guarantee content preservation (i.e., $I(\mathcal{Z};G|Z) \approx 0$ isn't guaranteed).
  • Proposes data augmentation by creating synthetic pairs $(Z, G^c)$ and using a mix-and-match loss $L_{\text{mm}}$ (Eq. 10).
  • Theorem 3.5: Shows training with $L_{\text{mm}}$ ensures $\mathcal{Z}$ is disentangled from $G$ conditional on $Z$, leading to editability that preserves content ($(\epsilon, \hat{\psi}, p_{X|Z})$-editable).
• Latent Subspace Models (LSMs):
  • Considers a specific case where $X = A_Z Z + A_G G$, with $A_Z, A_G$ defining orthogonal subspaces. Disentanglement here means recovering these subspaces. LSMs inherently avoid content distortion.
  • Proposes a specific regularized score matching loss $L^{\lambda_r}$ for LSMs (Eq. 12), featuring a style guidance regularization term $L_r$ that encourages maximizing the style contribution to the score. The architecture uses separate pathways for content and style scores (Figure 2).
  • Theorem 3.7: Shows minimizing $L^{\lambda_r}$ identifies the correct content and style subspaces.
  • Theorem 3.8: Analyzes gradient flow dynamics, showing convergence to the correct subspaces in the infinite-width limit when $\lambda_r=3$.
  • Theorem 3.9: Provides finite-sample guarantees, bounding subspace recovery error, disentanglement ($I(\mathcal{Z};G)$), and editability in terms of sample size $n$ and data dimension $d_X$.

Implementation Considerations:

• Architecture: The general framework uses a standard conditional DM architecture but adds an encoder $z_{\phi}$ to create the bottleneck $\mathcal{Z}$. For LSMs, a dual-encoder score network (Figure 2) is proposed, separating content and style processing. U-Net architectures are used for image/speech experiments.
• Training Objective: The key is the regularization. For the general case, $L^{\gamma,\rho}_c$ (Eq. 7) requires estimating mutual information, which can be challenging. The experiments use a simpler proxy inspired by the LSM case: $L_{c}^{\lambda_r}$ (Eq. 16) adds a term where the score is computed with the content bottleneck zeroed out, weighted by $\lambda_r$. For LSMs, the specific objective $L^{\lambda_r}$ (Eq. 12) including $L_0, L_b, L_r$ is used.
• Hyperparameters: The style guidance weight $\lambda_r$ (or $\gamma, \rho$ in the general theory) is crucial for achieving disentanglement, as shown theoretically and empirically. The diffusion time $T$ and noise schedule also play roles outlined in the theory.
• Data Augmentation: For tasks sensitive to content distortion, the mix-and-match loss (Eq. 10) requires generating synthetic samples by swapping styles, assuming an inductive bias allows this. Using multiple target speakers/styles in training acts as implicit data augmentation (validated in speech experiments).
• Computational Cost: Standard DM training costs apply. The added bottleneck encoder $z_{\phi}$ or the dual-path LSM architecture adds moderate overhead. Mutual information estimation (if implementing Eq. 7 directly) could add significant cost.

Experimental Findings:

• Synthetic (LSGMM): Successfully recovered latent subspaces. Subspace error decreased with larger style guidance weight $\lambda_r$ and more samples $n$, aligning with Theorems 3.8 and 3.9.
• Image (MNIST Colorization, CIFAR-10 Denoising): Showed that the style guidance regularization ($\lambda_r > 0$) was essential for disentangling content (digit/object shape) from style (color/noise) in an unsupervised setting (no paired data). Quantitative metrics (MSE, PSNR, SSIM, LPIPS) improved significantly with appropriate $\lambda_r$.
• Speech (Voice Conversion for Emotion Recognition): Using a DM-based voice conversion method (DiffVC) to adapt features for emotion classification (content: emotion, style: speaker) significantly outperformed baseline and other VC methods on IEMOCAP. Performance improved with more target speakers, supporting the theory on data augmentation (Theorem 3.5).

Conclusion:

The paper provides a theoretical basis for understanding and achieving disentanglement using diffusion models. It introduces regularized training objectives (notably style guidance) and analyzes their effectiveness, particularly for LSMs. The framework successfully connects theory to practice, demonstrating improved disentanglement and downstream task performance in image and speech domains through theory-inspired training strategies like regularization and data augmentation.