Title: Can Diffusion Models Disentangle? A Theoretical Perspective
Venue: NeurIPS 2025
Accepted: yes
Keywords: diffusion models

disentanglement

learning theory
Abstract: This paper presents a novel theoretical framework for understanding how diffusion models can learn disentangled representations with commonly used weak supervision such as partial labels and multiple views. Within this framework, we establish identifiability conditions for diffusion models to disentangle latent variable models with \emph{stochastic}, \emph{non-invertible} mixing processes. We also prove \emph{finite-sample global convergence} for diffusion models to disentangle independent subspace models. To validate our theory, we conduct extensive disentanglement experiments on subspace recovery in latent subspace Gaussian mixture models, image colorization, denoising, and voice conversion for speech classification. Our experiments show that training strategies inspired by our theory, such as style guidance regularization, consistently enhance disentanglement performance.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a theoretical framework for disentangling content and style in diffusion models. The authors first introduce two information-theoretic concepts called ($\\epsilon, \\nu$)-disentanglement and $\\epsilon$-editability. The former states that the restored content and style are approximately statistically independent, while the letter is satisfied if a change of style leads to a data distribution with unchanged content. Subsequently, two settings (access to the ground truth style and multiple views with shared content) and technical conditions are presented under which disentangling and editability can theoretically be achieved.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n* The paper adresses the problem of disentanglement in diffusion models, which is an important topic in generative modelling.\n* The paper provides a novel view on disentanglement from the lense of information theory\n\n**Weaknesses:**\nAt first glance, the theoretical results seem plausible, so I tried to follow the theoretical derivations in the appendix, but due to time constraints I could only look at a small part, as the derivations are quite extensive. Some equations seem to come out of nowhere (see questions below). So I have some concerns about the validity, but it is possible I didn't understand some parts.\n* The experimental part is difficult to follow. It is not clear to me from the short description how the datasets look like and how the results were obtained. The paper would benefit greatly from additional clarifications of the experimental setups.\n* There seems to be a gap between the theoretical contribution of the paper and its empirical validation. The presented theorems are based on the losses in Equation 7, 8, and 11, but the experiments are based on the loss in equation 13 without sufficient explanation how it relates.\n* It would improve readability of the appendix if theorems and lemmas were restated before the proofs. Otherwise one needs to switch back and forth quite a lot.\n\nQUESTIONS:\n* In Equation 7, what forces $I(\\hat{Z},\\hat{G};X)=I(Z,G;X)$ as $\\delta \\to 0$? Why can $\\hat{Z}$ not just be constant? \n* Why is the ground-truth decoder/generator $f$ in the score function in line 1122 and line 1204? Where is this equation for the score function coming from?\n* In line 1077 ,where is $C_2$ used?\n* How is the estimated content, $\\hat{Z}$, used in equation 13 for the experiments?\n* Why is the background in the CIFAR10 colorization task in the input always black? Shouldn't this also work with other background colors in the input?\n* How do ($\\epsilon, \\nu$)-disentanglement and $\\epsilon$-editability relate to deterministic formalizations of disentanglement? That is, there exist functions $h_1$, $h_2$ such that $\\hat{Z}=h_1(Z)$ and $\\hat{G}=h_2(G)$.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 1, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the disentanglement ability of diffusion models. The authors provide key theoretical insights, showing that under certain conditions with partial supervision or multi-view inputs, diffusion models can disentangle content and style latent variables. Additionally, in the special case of independent subspaces, they introduce a novel score loss and prove global convergence results with finite sample . These theoretical contributions are supported by numerical experiments, validating the convergence rate under the independent subspace assumption. The authors further validate the general disentanglement theorems on two real-world image datasets and in a voice conversion (VC) adaptation task.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n1.\tTheoretical contributions are solid and well-formulated.\n\n2.\tThe paper investigates an interesting problem, offering a novel perspective on disentanglement with diffusion models under certain supervision and assumptions.\n\n3.\tEven though the whole paper is notation-heavy, the writing style is accessible and visually engaging. The use of colored highlights and intuitive text boxes improves readability, and the examples in the main paper help understand key definitions.\n\n\n### Weaknesses\n\n1.\tThe connection between the multi-view setup and the independent subspace assumption is not clearly established for me. In my understanding, these are somewhat orthogonal concepts: independent subspaces do not necessarily imply a separation into content and style, and vice versa.\n\n2.\tSome of the assumptions made in this paper are quite strong and may not hold in real-world scenarios, potentially limiting the practical applicability of the theoretical findings. For instance, in the context of content disentanglement, the assumption that the style function is known is hard to justify for two main reasons:\n\n(1) Precisely identifying which part of the data corresponds to the style or changing factors is overly idealized and rarely feasible in practice.\n\n(2) Knowing the exact style value for all training samples is extremely challenging and unrealistic in most settings.\n\nAdditional assumptions that I find restrictive are listed in the question section.\n\nQUESTIONS:\n1. Does Definition 4.3 imply that the observation X is a linear combination of latent variables Z and G? If so, wouldn’t that be a rather strong assumption for real-world datasets?\n\n2. In the multi-view disentanglement setting, is it implicitly assumed that X and the concatenated latent variables (Z,G) have the same dimensionality? If so, could this constrain the applicability of the model?\n\n3. I’m having trouble understanding how the example: \"a speaker embedding extracted from a pre-trained speaker recognition model\", which is used to illustrate a case where the style value is known. Could you explain more about how this is connected to \"knowing the style function/value\"? Is the speaker embedding considered the exact value of the style variable G? Or does using a pre-trained model imply that we fully know the style function, or are we just estimating it?\n\nLIMITATIONS:\nAlthough the paper is theoretically solid and presents three key theorems, I find it challenging to build a clear logical connection or progression among them, particularly between the final theorem and the first two. It would be helpful if the authors could provide more explanation on how these results build upon one another or fit into a cohesive framework. Additionally, some of the assumptions made in the theoretical analysis are quite strong. I list them all in both the weakness and question parts and would appreciate further discussion from the authors on how these assumptions might be relaxed or addressed.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a theoretical framework for learning disentangled representations using conditional diffusion models under weak supervision conditions, such as partial labels and multiple views. While most prior work on identifiable representation learning has focused on deterministic, invertible mixing processes, diffusion models inherently involve stochastic, non-invertible mixing processes. To address this, the authors propose approximate disentanglement within conditional diffusion models under partial supervision or multi-view scenarios, supported by theoretical guarantees derived from Lipschitz continuity assumptions. The theoretical results are further validated through empirical experiments involving Gaussian mixture models, MNIST colorization, and CIFAR-10 image denoising, demonstrating effective disentanglement capabilities of the diffusion models.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\nThe paper is well-written and easy to follow. Formulating a definition of formal disentanglement for stochastic and non-invertible mappings of latent variables is a valuable contribution to the field. This paper introduces a clearly defined formulation of approximate disentanglement through the concepts of $(\\epsilon,\\nu)$-disentanglement and $\\epsilon$-editability, addressing an important theoretical gap.\n\n\n**Weaknesses**\n\nA major concern is in the empirical justification of the proposed framework. Although the proposed theoretical foundation presented is concrete, the connection to empirical evidence is somewhat weak. For example, how does Equation (13) derived from Equation (11), why does it lack balancing loss, how can Equation (13) apply to datasets where it does not follows independent subspace model (ISM), lack of experiments on multi-view disentanglement, etc. Please refer to the Question Section below. Weak connection to the empirical study makes it challenging to assess the empirical validity and overall contribution of the proposed theoretical framework.\n\nQUESTIONS:\n- In Equation (10), what is $U$ and $V$? They are not explicitly defined in the manuscript. As Equation (11) only optimizes for $\\theta$ and not $U$ or $V$, it seems they are not learnable components. Are they some kind of pre-defined matrix?\n\n- Could the authors clarify the meaning of $n$ in Equation (11)?\n\n- In Line 235, the paper mentions that “The style guidance regularizer encourages style separation by reducing the mutual information between the content encoder and $X$.” However, it is unclear why score prediction using only style information (i.e., $Vs_G(g(x),t)-s^*(x,t)$) necessarily reduces mutual information between the content encoder and $X$. Could the authors clarify if it is possible for the content encoder to still capture excessive information about the style?\n\n- Could the authors elaborate more on the role of balancing loss in Equation (11)? Although the full derivation and proof are presented in Appendix, it was personally hard to follow the full derivation. Could the authors provide intuition on how this term avoids poor local minima? \n\n- In the experimental section, it is mentioned that Equation (13) is used for training on MNIST and CIFAR-10 as well. However, it is not clear how Equation (13) is derived from Equation (11). Firstly, it lacks balancing loss. Does it imply that balancing loss is a redundant term from the formulation? Additionally, since Equation (11) relies on the assumption of the independent subspace model (ISM), could the authors discuss the validity of applying Equation (13) to datasets like MNIST and CIFAR-10, where this assumption may not hold?\n\n- Although multi-view disentanglement is highlighted as a core contribution, relevant experimental validation is missing. Could the authors provide empirical evidence demonstrating multi-view disentanglement?\n\n- In image disentanglement experiment, could the authors provide comparison to related work (e.g., [1,2,3])? It would improve overall understanding if the authors provide comparison to previous work and provide analysis. \n\n- A wrong image is inserted in Figure 6(d) in the Appendix. \n\n[1] Chen et al., Weakly supervised disentanglement with guarantees, in AAAI 2020. \n\n[2] Shu et al., Weakly supervised disentanglement with guarantees, in ICLR 2020. \n\n[3] Locatello et al., Weakly supervised disentanglement without compromises, in ICML 2020.\n\nLIMITATIONS:\nyes, (although the authors do not allocate a section for indicating the limitation, it was written in the discussions throughout the paper)", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper establishes a theoretical framework for diffusion model based disentanglement under weak supervision. The authors address key challenges caused by the stochastic and non-invertible nature of DMs and propose two formal criteria for approximate disentanglement. They provide theoretical guarantees for both content-style disentanglement and multi-view disentanglement.  Experiments across synthetic, image, and speech data.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Well-Defined Metrics: Introduces new formal definitions for approximate disentanglement ((ε,ν)-disentanglement and ε-editability) suited to diffusion models.\n- Well-structured theory​​: Clear progression from general identifiability (Sec 4.1–4.2) to specialized ISM convergence (Sec 4.3).\n\nWeaknesses:\n- The ISM assumption may not hold for complex data. Experiments are mostly small-scale and controlled; there’s a lack of large-scale, diverse benchmarks that would test generalization in more realistic scenarios.\n- Limited novelty: The idea of employing mutual information-based regularization to promote disentanglement in diffusion models has been previously explored, notably in DisDiff [1].\n- The work lacks empirical comparisons against existing content-style disentanglement approaches, making it difficult to assess the relative effectiveness of the proposed method.\n- The title may misslead the readers, since the paper mostly focuses on content-style disentanglement, which is different from the general disentanglement [1,2,3]\n\n[1]  Yang, Tao, et al. \"DisDiff: Unsupervised Disentanglement of Diffusion Probabilistic Models.\" Advances in Neural Information Processing Systems 36 (2023).\n\n[2] Kim, Hyunjik, and Andriy Mnih. \"Disentangling by factorising.\" International conference on machine learning. PMLR, 2018.\n\n[3] Chen, Xi, et al. \"Infogan: Interpretable representation learning by information maximizing generative adversarial nets.\" Advances in neural information processing systems 29 (2016)\n\nQUESTIONS:\nDose the method works on CelebA? may qualiative results and quantitative results compare with some recent methods [1,2] will make this work more solid.\n\n[1] Ren, Xuanchi, et al. \"Rethinking content and style: exploring bias for unsupervised disentanglement.\" Proceedings of the IEEE/CVF International Conference on Computer Vision. 2021.\n[2] Wang, Zhizhong, Lei Zhao, and Wei Xing. \"Stylediffusion: Controllable disentangled style transfer via diffusion models.\" Proceedings of the IEEE/CVF International Conference on Computer Vision. 2023.\n\nLIMITATIONS:\nSee weakness.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 1, "ORIGINALITY": 1, "DATE": "2025-07-07"}