Title: When and how can inexact generative models still sample from the data manifold?
Venue: NeurIPS 2025
Accepted: yes
Keywords: Generative models; machine learning theory; dynamical systems theory;
Abstract: A curious phenomenon observed in some dynamical generative models is the following: despite learning errors in the score function or the drift vector field, the generated samples appear to shift \emph{along} the support of the data distribution but not \emph{away} from it. In this work, we investigate this phenomenon of \emph{robustness of the support} by taking a dynamical systems approach on the generating stochastic/deterministic process. Our perturbation analysis of the probability flow reveals that infinitesimal learning errors cause the predicted density to be different from the target density only on the data manifold for a wide class of generative models. Further, what is the dynamical mechanism that leads to the robustness of the support? We show that the alignment of the top Lyapunov vectors (most sensitive infinitesimal perturbation directions) with the tangent spaces along the boundary of the data manifold leads to robustness and prove a sufficient condition on the dynamics of the generating process to achieve this alignment. Moreover, the alignment condition is efficient to compute and, in practice, for robust generative models, automatically leads to accurate estimates of the tangent bundle of the data manifold. Using a finite-time linear perturbation analysis on samples paths as well as probability flows, our work complements and extends existing works on obtaining theoretical guarantees for generative models from a stochastic analysis, statistical learning and uncertainty quantification points of view. Our results apply across different dynamical generative models, such as conditional flow-matching and score-based generative models, and for different target distributions that may or may not satisfy the manifold hypothesis.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper explores the surprising empirical observation that generative models GMs often generate samples that remain close to the support of the data distribution—even when the learned score function is inaccurate. The authors investigate this phenomenon through a dynamical systems lens, modeling the generative process as a random dynamical system and conducting a perturbation analysis on both the probability flows and sample paths.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- this is an excellent high-quality paper with deep general insights into the robustness and behavior of the main generative models used today.\n- the results are extremely general, applying to all practical implementations of GMs\n- the results are clear, extremely significant, and original\n- theoretical analysis is rigorous, with careful derivation of perturbation response fields and a novel use of finite-time Lyapunov theory.\n- sufficient conditions for support robustness are justified and grounded in both empirical observation and mathematical derivation.\n- The paper is well-structured, guiding the reader through a difficult topic with illustrative examples and clear definitions.\n- It connects geometry, dynamical systems, and deep generative models in a novel way that could influence future design and evaluation.\n\nweaknesses:\n- Concrete suggestions to improve to GMs are not mentioned, even vague suggestions would be welcome.\n\nQUESTIONS:\nThe current theory provides sufficient conditions for robustness. Are there indications of necessary conditions, or empirical observations of counterexamples that satisfy some but not all conditions?\n\nCould the alignment property be enforced or encouraged during training via a regularization term?\n\nYour results are framed for infinitesimal or small perturbations. How does the theory behave when learning errors are non-negligible?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work attempts to theoretically explain the robust performance of generative models in the presence of approximation error in the score function. The main message of the paper is the following: even under the approximation error, the generated samples drift only in the directions along the support, not away from it. As a result, the generated samples are not too different from the target samples. The main analysis of the work relies on the perturbation theory and analyzes the properties of the top Lyapunov vector (the most sensitive perturbation direction). I think this is an important problem to study and requires future research. However, I find the presentation of the paper extremely hard to follow. I elaborate on this in the later section.\n\nSTRENGTHS AND WEAKNESSES:\n# Strength\n\nThe paper addresses an important problem that might broaden our theoretical understanding of the contemporary generative models. The paper introduces a novel mathematical framework for analyzing the dynamics of the generated samples using perturbation theory. I am not an expert in perturbation theory, but it seems that the present framework might be helpful in future research in this direction. \n\n# Weakness\nI find the presentation of the paper extremely hard ot follow. \n\n- The paper is very notation-heavy, and it is hard to follow the claims for a non-expert reader. I encourage the authors to add some sort of table for notations.\n\n- Line 180 - 185: The phenomenon regarding the top Lyapunov vector explained in the aforementioned lines is not quite clear. It will be helpful if the authors provide more figures to explain the details of the dynamics. More importantly, definite examples supporting the claims would be helpful for the readers. For example, verifying the claims in simple settings, such as the DDP model with a mixture of Gaussian distributions as the target distribution,  is encouraged for better readability.\n\n- The main results of the paper are hard to parse. For example, the implication of Proposition 4.1 is not clear. Also, what is \"margin\" when you write $c = O(1/ margin)$ in Line 213? I could not find any definition.\n\nQUESTIONS:\n- What is the main implication of Theorem 4.3? It would be helpful if the authors could provide more figures with explanations.\n\n- Line 275: What is the meaning of \" aligned and convergent generative model\"? An explicit definition will be helpful to the readers. \n\n- Also, how to verify the alignment of a model in practice?\n\n- Lemma 4.4: What is the meaning of regular? An explicit definition is \"alignment property,\" and \"regular\" is helpful.\n\n- \n\nline 311: \"An SGM typically satisfies this condition.\":  Is that trivial? A short argument might be helpful.\n\nLIMITATIONS:\nAs of now, the scope of the work is very theoretical and limited. Further experimental verifications are necessary to gauge the impact of the work.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 1, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, authors study the robust behavior of dynamical generative models. \n\nAuthors first show that even with erroneous perturbation, the infinitesimal change of the target density is only supported where the target density is supported. To explore the manifold hypothesis, the authors introduce the notion of alignment: the principal directions of sample deformation—captured by the leading Lyapunov vectors—are orthogonal to the target score at the boundary of the data manifold. A sufficient condition for alignment is also derived for generative models. \n\nBoth analyses explain the robustness of dynamical generative models and why they can reliably produce samples on the target manifold even with statistical estimation errors. \n\nSome empirical results are performed on toy and real-world datasets, validating the theoretical claim.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n\n1. The manifold hypothesis (Song and Emron., 2019, [45, 14] in the manuscript) is an important conjecture and is believed to be the reason that dynamical generative model work so well comparing to other generative methods (e.g. GAN). Mathematical analysis on this subject will have significant impact on down-stream applications. \n2. Authors' analysis assumes a very generic framework of dynamical generative models; thus, the results are applicable to a wide variety of drift based generative models (such as diffusion models and CFM). \n3. The sufficient conditions of manifold alignment theorem have intuitive interpretations and can be computationally validated even on high-dimensional data (Figure 2). \n\nSong and Emron, Generative Modeling by Estimating Gradients of the Data Distribution, 2019\n\nWeaknesses and Comments: \n\nOverall, this paper is technically dense and could benefit from additional examples illustrating these mathematical concepts. \n \nThe first result (Section 3) is not very well explained. Authors could expand the paragraph after eq. 2, explaining the result and its implications in a more intuitive manner (similar to line 274 - 288). For example, why does the eq. 2 show that generative model sticks to the target support? It is not immediately clear as the rho_tau(x) in eq.2 is not target density p_data. \n\nSome elements in the sufficient conditions in Theorem 4.3 are very abstract, (e.g., nabla_t,d v_t,d), and are not easily observable from Figure 1. Could authors create a new plot to show how do these quantities behave on the boundary of the manifold? \n\nAfter Theorem 4.3. I recommend introduce some toy examples (e.g. diffusion generative model learning a target density on a sphere) and show the validity of the sufficient conditions using this simple example. \n\nTo a general reader who may not have extensive experience on dynamical generative model, some terminologies may be hard to understand. For example, \"compressive\", \"expansive phase\", \"superstable\", \"non-singular target density\" etc.\n\nQUESTIONS:\nIf the author could kindly answer the following questions, it would greatly helpful: \n\n- > Intuitively this means that the generating dynamics may include an expansive phase and the vector field vt is a\n uniform “attracting force” at the end.\n\nCould authors explain how this intuition is derived from the sufficient conditions listed in Theorem 4.3? Specifically, it is not clear to me which part of the sufficient condition implies that we could have an expansive phase. \n\n- What does the condition (ii) mean in Theorem 4.3? Could we check if this condition is satisfied in Figure 1?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

comments: SUMMARY:
This paper investigates a key empirical observation in dynamical generative models, where errors in the learned vector field (score) often shift generated samples along the data manifold but not off it. The authors propose a theoretical explanation for this phenomenon via a dynamical systems perspective, focusing on finite-time Lyapunov vectors.

STRENGTHS AND WEAKNESSES:
The paper provides rigorous and detailed mathematical analysis, bridging generative modeling with classical dynamical systems theory, particularly via Lyapunov analysis and transfer operators. However, the paper focuses on sufficient but not necessary conditions, leaving the full characterization of robustness open.

QUESTIONS:
1. I am a bit confused about the 'tangent perturbation' at line 120. Does it mean the perturbation along or away from the data manifold? To study the robustness of support, I believe we should consider the perturbation pushing the samples away from the manifold, but 'tangent' sounds like making the samples stick to the manifold. Same question for the last column of Figure 1: what does it mean by shifted 'tangent' to the data manifold? Also would you please clarify what it means by 'perturbations along sample paths' at line 157?

2. At line 68, the author states '... as the perturbation size tends to 0', which looks like indicating an asymptotic result. But I would expect the result like 'as the perturbation size $\epsilon$ is sufficiently small'. Please correct me if I have a misunderstanding.

3. The notation of $u$ defined in section 3 is a bit confusing, as $u$ is a vector field while $u(\rho_\tau)$ is a scalar. I would suggest using different notations.

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-03