Title: Neural Entropy
Venue: NeurIPS 2025
Accepted: yes
Keywords: Diffusion models

thermodynamics

information theory

statistical mechanics
Abstract: We explore the connection between deep learning and information theory through the paradigm of diffusion models. A diffusion model converts noise into structured data by reinstating, imperfectly, information that is erased when data was diffused to noise. This information is stored in a neural network during training. We quantify this information by introducing a measure called \textit{neural entropy}, which is related to the total entropy produced by diffusion. Neural entropy is a function of not just the data distribution, but also the diffusive process itself. Measurements of neural entropy on a few simple image diffusion models reveal that they are extremely efficient at compressing large ensembles of structured data.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work introduces the concept of neural entropy, to quantify the amount of information stored in a neural network. The focus of this work is on diffusion-based generative models, and thus the “score network” or “denoising network” that is typically used to learn how to reverse the diffusion dynamics. Throughout a very didactical derivation, the authors argue that diffusion models are very effective in storing information, especially when dealing with large amounts of structured data: this claim is supported by mathematical derivations as well as simple experiments both on synthetic data and on simple image datasets.\n\nUsing neural entropy, the authors show that the growth of neural entropy with the number of training samples in diffusion models obeys to logarithmic scaling, suggesting that the marginal information gained per sample decreases as approximately $1/N$. Moreover, they also show that there is some sort of “speed limit” in the sense that diffusion models that employ “tricks” to make them faster, also need to store more information for an accurate reversal process.\n\nFurthermore, this work introduces the entropy-matching objective, as a way to tighten the KL divergence between the data distribution and its parametric approximation achieved with a “denoising network”. This is in contrast to a score-matching objective, for which a neural entropy expression is conceivable, but hard to interpret, especially in the case of a variance preserving process. Instead, the entropy-matching objective has a clearer interpretation in terms of neural entropy.\n\nFinally, this article presents a series of simple experiments to validate numerically the main theory of the paper: empirical lines match the predictions of the ideal ones, supporting the claims of the authors.\n\nSTRENGTHS AND WEAKNESSES:\n* Strengths:\n  * This is a very easy to read article, very pedagogic and clear. It has many examples, and begins with simple processes inspired by established results in physics, but continue with more elaborate and realistic stochastic processes that are used in modern generative models. \n  * This work presents a very intuitive explanation for the amount of information stored into a network used to reverse the dynamics of diffusion processes. The relation between information storage and the structure of the data, the number of training samples, and the particular form of the stochastic process used is important and could be used to better engineer diffusion-based generative models\n  * The authors introduce the entropy-matching objective, which is easy to implement and optimize, similarly to flow-matching objectives, although care needs to be taken to select appropriate measures for numerical stability\n  * I liked the discussion and perspectives in the conclusion, aiming at answering the question about the application of the proposed neural entropy measure to other generative models, such as language models. The connections to diffusion-based LLMs is pertinent and interesting\n\n* Weaknesses:\n  * I’m lucky enough to have sufficient background on the literature required to understand this paper, but for a generic reader, this work might be somehow difficult to digest, as it is not self-contained. The (long) appendix does a good job in providing background material and proofs, but it requires a particular effort to “put all pieces together”\n  * The practical application of neural entropy to image diffusion models is hindered by some instabilities and calls for more sophisticated diffusion-based methods to avoid divergence\n  * The authors did not put a particular effort in translating the methodological results presented in this work into actionable information for practitioners who might use entropy-matching, and neural entropy to design novel (and better) image diffusion models\n\nQUESTIONS:\n* What are the main practical implications of this work? Can you illustrate how your ideas can help designing new diffusion-based generative models? This kind of discussion could help broaden the scope and audience of your work\n\n* I think the problems with score-based diffusion models, having a drift term that is “repulsive”, is important. The discussion in the appendix and in particular Figure 9, right-hand side, points at a possible mismatch of the neural entropy metric for score-based models. Can you help me clarify wether this implies that the proposed method to measure information stored in a score-matching based model is limited, or if there are any important limitations to score-matching models due to this drift term?\n\n* Recently, there has been a significant increase of work that use diffusion models, as well as other generative models to compute information theoretic quantities such as Entropy and Mutual Information. In those cases, the key objective is to compute the entropy of a given data distribution, using score networks for example, or to compute the mutual information between two data distributions, again using score networks for example. Do you think these works are relevant to some extent and could be discussed as related work?\n\n[1] Butakov et. al., “Mutual Information Estimation via Normalizing Flows”, https://arxiv.org/abs/2403.02187 \n\n[2] Franzese et. al, “Mutual Information Neural Diffusion Estimation”, https://arxiv.org/abs/2310.09031\n\n[3] Kholkin et. al, “InfoBridge: Mutual Information estimation via Bridge Matching”, https://arxiv.org/abs/2502.01383\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper defines and analyses a measure called neural entropy. The definition of neural entropy is justified via concepts of information theory and thermodynamics applied to random walk processes and related to diffusion models in machine learning. The analysis of this measure is performed empirically in both a toy controlled setup, where the ground truth entropy is known, and an uncontrolled setup, using two common datasets in the machine learning community.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n- The paper clearly states, at multiple places, the limitations of the neural entropy measure and the experiments performed.\n\n- The analysis performed gives several important insights on neural networks, namely on their capability to store information and the rate at which they learn it.\n\n- The analysis of neural entropy on a controllable multivariate Gaussian case gives clear and interpretable results.\n\n- The conclusion on the success of the variance preserving scheme is interesting.\n\n## Weaknesses\n\n- I find the definition of neural entropy a bit confusing. Equation (11) defines the ideal neural entropy as the total entropy, while equation (18) defines the “actual” neural entropy as the subpart of the total entropy that is due to the diffusion model formulation. This shift in definitions creates a confusion on the interpretation of the measure.\n\nQUESTIONS:\n## Questions\n\n1. Section 6, paragraph “Transport experiments”: Good point to limit the inductive biases here, but what about the other effects on the neural entropy such as the size of the network, the choice of the optimisation algorithm, the batch size, the number of samples per epoch (or did you resample your Gaussians at each epoch as if an infinite amount of data was available?), etc.? Did you notice any interesting behaviour or was the neural entropy stable with respect to such hyperparameters? Sorry if I missed this in the text.\n\n1. Did you try your method on a larger and much more complicated dataset such as ImageNet? As I acknowledge that one could always try more, it is not a weakness strictly speaking. Nevertheless, it would be interesting to check whether, at a certain point, there is a limit to the inductive biases of the network which would manifest via the neural entropy.\n\n1. Section 6, paragraph “Storage experiments”: Do you have any quantitative argument to justify that the model is indeed not memorising?\n\n## Suggestions\n\n- Section 6, paragraph “Transport experiments”, line 4: There may be a typo with the reference to Eq. (15), which I think should be Eq. (17).\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper explores the interplay between deep learning and information theory through the lens of diffusion models. It proposes a novel metric, neural entropy, to quantify the amount of information retained within the neural network during training. This measure captures aspects of both the underlying data distribution and the dynamics of the diffusion process. Empirical evaluations on image diffusion tasks demonstrate that these models achieve remarkable efficiency in compressing structured data.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\n1. The paper is well-motivated and makes a meaningful contribution to the theoretical understanding of diffusion model training.\n2. It is well-written and presents extensive theoretical analysis, supported by empirical results that validate the proposed theory.\n\nWeaknesses\n\n1. The relationship between network size and neural entropy requires further consideration. The paper would benefit from additional theoretical analysis and experiments addressing this relationship. Referring to related studies, such as [A], could strengthen this aspect.\n2. Including a discussion on generalization could strengthen the impact of this study. In particular, it would be valuable to investigate whether the model tends to overfit or generalize when training proceeds with new data after entropy saturation. Such an analysis would provide deeper insight into the relationship between entropy dynamics and generalization behavior.\n\n[A] Morris, John X., et al. \"How much do language models memorize?.\" arXiv preprint arXiv:2505.24832 (2025).\n\nQUESTIONS:\nI would appreciate a detailed and thoughtful response to the concerns raised in the Weaknesses section. If these concerns are adequately addressed, I would be willing to reconsider my evaluation and potentially adjust the score in a more positive direction.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper aims to measure the information that a trained diffusion model stores about its training data. The main result of the paper is that this information is proportional to $\\log n$ where $n$ is the training dataset size, rather than $n$, as would be the case for a memorizing network sampling from the empirical training data distribution.\n\nSTRENGTHS AND WEAKNESSES:\nThis paper tackles a fundamental question in diffusion generative modeling, and its results are very intriguing. However, in its current state, I find it difficult to understand, especially as a non-physicist. \n\nSome of the physics jargon is difficult to understand for non-physicists: e.g., top of page 2, \"the entropy of the phase space density over the neural network’s internal microstates\" does not make any sense to me. Referencing \"the total entropy produced by a diffusion\" without motivating its expression or the Maxwell demon without briefly introducing it is also not appropriate given the audience of NeurIPS. I also don't understand what it means for the marginal distribution of a particle evolving under an SDE to itself fluctuates: by definition the distribution is deterministic, and evolves under an ODE (Fokker-Planck). It seems that what is meant is the empirical distribution of the $M$ walkers, but in that case it cannot be equal to $p_{\\mathrm{d}}$ nor $p_{\\mathrm{eq}}$.\n\nI end up quite lost in the notation, especially given that some objects are never explicited (e.g., the kernel $h_\\star$: does it admit a closed-form equation?) or redefined multiple times. For instance, during the first reading I got confused between $S_{\\mathrm{tot}}$, $\\hat S_{\\mathrm{NN}}$, and $S_{\\mathrm{NN}}$, or $s_\\theta$ and $e_\\theta$, or $p_0$, $p_{\\mathrm{eq}}$, and $p_{b_+}$. These quantities are often introduced early in the text, not used for a while, then mentioned later without reminding the reader what they stand for.\n\nThe main issue is that after reading the main text and the appendix twice, and spending some time thinking about it, I am not confident at all about what is measured in Figure 1, the central result of the paper. More specifically, I don't see how or why the proposed definition of $S_{\\mathrm{tot}}$, and even less so $S_{\\mathrm{NN}}$, should coincide with the \"information load\" on the network. First, in the so-called \"variance-exploding\" SDE (where $b_+ = 0$), $S_{\\mathrm{tot}}$ is simply equal to $H(p_{\\mathrm{eq}})$ - $H(p_{\\mathrm{d}})$. This does not seem to me to be a reasonable estimate of information \"needed to reverse the diffusion process\" in that case. Second, after several pages of careful motivation and explanation of $S_{\\mathrm{tot}}$, $S_{\\mathrm{NN}}$ is introduced instead without explanation. As a result, I have no intuition what it measures.\n\nFinally, the following reference might be relevant to the authors, as it derives several information-theoretic relations in diffusions that are very related to those in this paper:\n> Kong, X., Brekelmans, R., & Steeg, G. V. (2023). Information-theoretic diffusion. arXiv preprint arXiv:2302.03792.\n\nTo summarize:\n- Strengths: the paper studies an important topic, and the claimed results are very significant (though I cannot evaluate them fully due to my limited understanding of the paper)\n- Weaknesses: the style of writing is hard to follow, and the final definition of $S_{\\mathrm{NN}}$, central to the paper, is not motivated enough (why should it measure the amount of information about the training set stored by the diffusion model?)\n\nQUESTIONS:\nWhat is $S_{\\mathrm{NN}}$ measuring? Why would it be the correct measure of information stored in the network? Is it still valid for an untrained network? It seems it can be made arbitrary large for arbitrary networks. What do the equations look like in the variance exploding case ($b_+ = 0$), do the authors still think of them as valid in that setting? Why are the equations not invariant to time reparameterization? Simply rescaling time should not fundamentally change the information needed to store the training distribution.\n\nIn my current state of understanding of the paper, I cannot recommend acceptance, but I am willing to improve my score if the authors convincingly demonstrate the well-foundedness of their entropy measure.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 1, "CLARITY": 1, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-03"}