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EGEA-DM: Eigenvalue-Guided Explainable and Accelerated Diffusion Model |
Soundness: 3: good
Presentation: 4: excellent
Contribution: 2: fair
Rating: 4: marginally below the acceptance threshold
Confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. |
This paper introduces EGEA-DM, a framework intended to address the high computational costs, slow convergence, and perceived lack of theoretical interpretability in existing diffusion models. The central thesis is the application of ergodic theory to the generative diffusion process. Specifically, the authors propose a formal connection between the convergence rate of the forward diffusion process and the principal eigenvalue ($\lambda_1$) of its corresponding L-generator. The framework's core mechanism involves modulating this principal eigenvalue by adjusting the coefficients $a(x)$ and $b(x)$ of the L-generator. The authors provide a theoretical argument (Theorem 2) and empirical results (on CIFAR-10 and CelebA-HQ-64) to support their main claim: a larger principal eigenvalue $\lambda_1$ leads to an exponentially faster convergence to the stationary distribution. This, in turn, allows the model to reach convergence in fewer forward steps, significantly reducing the computational overhead for both training and sampling. The method is also presented as a plug-and-play module compatible with existing samplers like DPM-Solver.
- The primary strength of this work is its effort to ground the problem of diffusion model acceleration in established mathematical theory. While the underlying concepts of spectral gap convergence for 1D diffusions (e.g., from Chen 2005/2012, Bobrowski 2008) are not new, their application as a design principle for accelerating generative models is a valuable contribution.
- The paper does not just posit a theoretical relationship. It provides a operational method for implementing its core idea by adapting an iterative numerical estimation algorithm (from Chen 2012) to estimate the principal eigenvalue $\lambda_1$. This turns a purely theoretical quantity into an actionable component of the algorithm.
- The paper is generally well-written and logically structured. The narrative guides the reader from the ergodic theory, through the specifics of the L-generator and its eigenvalue, to the eventual experimental validation.
- The entire theoretical derivation (Preliminaries, Section 3.1) is explicitly simplified to a one-dimensional (1D) case for tractability. However, diffusion models for image generation operate in extremely high-dimensional spaces. The paper makes no serious attempt to bridge this enormous theoretical gap. Spectral gap analysis in high dimensions is notoriously more complex than in 1D and often requires additional structural assumptions. The authors do not discuss how, or even if, their 1D-derived insights generalize to high-dimensional, non-reversible, or anisotropic processes common in diffusion modeling.
- The proofs for the main theorems, such as Theorem 2, are relegated to the appendix and are overly brief. They lack the rigor and detail necessary for a reader to verify the claims or, more importantly, to understand the precise preconditions and scope under which these convergence bounds hold. This ambiguity undermines the paper's theoretical foundation.
- The paper claims to provide an "explainable" framework. However, "Guiding Principle II" (which states that higher polynomial orders for $a(x)$ and $b(x)$ lead to a larger $\lambda_1$) is presented as a purely empirical observation from numerical experiments. This is a heuristic, not an explanation. The paper fails to provide any theoretical justification for why this relationship holds, which runs counter to its own stated objective of improving interpretability.
Collectively, these issues suggest that the theoretical contribution of the paper is largely limited to restating classical diffusion results in a simplified 1D setting, without sufficient rigor or justification to claim novel theoretical insight.
The proposed method introduces an extra, non-trivial computational step: the iterative numerical estimation of $\lambda_1$. The appendix (C.2) claims this overhead is "entirely acceptable." However, the framework's performance seems to hinge critically on this value. How sensitive is the model's performance (e.g., final FID, optimal $T_{conv}$) to the accuracy of this numerical estimation? If the estimation error is, for example, 10% or 20%, does the principled acceleration break down or become suboptimal? |
Fully AI-generated |
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EGEA-DM: Eigenvalue-Guided Explainable and Accelerated Diffusion Model |
Soundness: 2: fair
Presentation: 1: poor
Contribution: 1: poor
Rating: 4: marginally below the acceptance threshold
Confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. |
The paper proposes EGEA-DM, which applies ergodic theory to diffusion models by controlling the principal eigenvalue of the L-generator to regulate forward diffusion speed. The authors claim this provides theoretical interpretability and enables faster training while maintaining generation quality. Experiments on CIFAR-10 and CelebA-HQ-64 show reduced training steps, though with mixed results on quality.
1. **Interesting theoretical angle.** Connecting the spectral gap of the L-generator to convergence rates is a neat idea, and framing diffusion acceleration through ergodic theory is relatively unexplored in the generative modeling literature.
2. **Rigorous mathematical development (in 1D).** Theorems 1-2 properly characterize uniqueness, ergodicity, and convergence rates for one-dimensional diffusion processes. The eigenvalue estimation procedure based on Chen (2012) is technically sound.
3. **Integration with existing samplers.** The framework works with DPM-Solver and DPM-Solver++, suggesting it could be modular.
### 1. The theory lives in 1D but the experiments are in high dimensions
This is the paper's fundamental problem. All theoretical guarantees (Theorems 1-2, convergence rates, eigenvalue analysis) assume one-dimensional processes where components evolve independently. But image generation requires modeling high-dimensional joint distributions with complex spatial dependencies.
The paper claims "without loss of generality" we can focus on 1D (line 59), but this is incorrect. For a 256-dimensional vector (e.g., 16×16 grayscale image), the spectral gap of 256 independent 1D processes tells you nothing about the mixing time of the joint distribution. The covariance structure, spatial correlations, and coupling between dimensions fundamentally change the convergence behavior.
There's no analysis of how the 1D eigenvalue relates to the high-dimensional process actually being run. This makes the entire theoretical framework disconnected from the experimental validation.
### 2. The practical recipe doesn't follow from the theory
Section 3.3's "guiding principles" for choosing L-generators feel ad-hoc:
- Principle I just says "satisfy the ergodicity conditions" (obvious)
- Principle II says "higher degree polynomials give bigger eigenvalues" (empirical observation, not a principle)
The paper never explains *why* polynomial forms are the right parameterization, or how to choose between the infinitely many (a,b) pairs that satisfy the convergence conditions. Table 4 shows that nonlinear generators behave unpredictably. Even with similar eigenvalues, you get wildly different Ddisc values (Appendix C.3, Table 7).
The guidance provided amounts to "try different polynomials and see what happens," which undermines the claim of interpretability and principled control.
### 3. Computational cost story is incomplete
The paper emphasizes training acceleration but glosses over the eigenvalue estimation overhead. In Appendix C.2, the authors mention "about two hours" for computing eigenvalues, but:
- This is for 1D. What about estimating properties of the high-dimensional process?
- How does this scale with dataset complexity or resolution?
- What's the total wall-clock time (estimation + training + sampling) compared to just training a baseline DDPM longer?
Table 1 shows training drops from 52h to 26h, but if I need 2+ hours of eigenvalue computation for every new (a,b) configuration I try, plus the overhead of determining the "optimal" range, the net savings become unclear. There's no end-to-end timing comparison.
### 4. Experiments are too limited
**Baselines:** The paper primarily compares against vanilla DDPM from 2020. For a paper submitted in 2025, missing comparisons include:
- DDIM (deterministic sampling, 2021)
- EDM, EDM2 (2022,2024)
- Modern flow matching methods (2023-2025)
**Datasets:** Only 32×32 CIFAR-10 and 64×64 CelebA. No experiments at 256×256 or higher resolutions where diffusion models are most impactful. No text-to-image, video, or other modalities.
### 5. Changing (a,b) changes the target distribution
This is subtle but important. When you modify a(x) and b(x), you don't just change the convergence speed—you change the stationary distribution π (Theorem 1). So comparing FID scores across different eigenvalue configurations isn't purely measuring the speed/quality tradeoff; you're potentially converging to different distributions.
The paper doesn't discuss this. Are the quality changes due to insufficient sampling steps, or because you've fundamentally altered what you're sampling from?
## Minor Issues
- **Writing quality:** Several grammatical errors ("egodic theory" in abstract, "beolw Theorem 1" line 168). Notation switches between $X_t$ and $Y_t$ confusingly.
- **$D_{disc}$ metric:** The convergence discrepancy metric behaves irregularly for nonlinear cases (Table 7) and seems unreliable as a stopping criterion. Its relationship to actual generation quality is unclear.
- **Figure 5:** Referenced as justification for polynomial choices but relegated to the appendix. The 3D surface plot is hard to interpret and doesn't provide clear design guidance.
- **Table 5:** Shows that very different (a,b) with similar eigenvalues give similar results. This contradicts the claim that eigenvalue is the dominant factor. if other properties of (a,b) matter equally, what's the advantage of the eigenvalue-centric view?
1. Can you provide a rigorous treatment of how 1D eigenvalues relate to high-dimensional convergence, or alternatively, show how to compute/estimate the spectral gap of the actual high-dimensional process?
2. What happens when you do compute-normalized comparisons (same total compute budget including eigenvalue estimation) against strong baselines?
3. How do you explain Table 5, where eigenvalue seems less important than other properties of the generator?
4. Can you clarify whether the quality differences come from speed/sampling tradeoffs or from changing the target stationary distribution? |
Fully AI-generated |
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EGEA-DM: Eigenvalue-Guided Explainable and Accelerated Diffusion Model |
Soundness: 3: good
Presentation: 3: good
Contribution: 3: good
Rating: 6: marginally above the acceptance threshold
Confidence: 3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked. |
This paper provided the theoretical understanding for the sampling rate of diffusion model via the egodic theory and proposed an efficient method called Eigenvalue-Guided Explainable and Accelerated Diffusion Model (EGEADM). EGEADM leverages the L-generator’s principal eigenvalue to explicitly control the sampling speed and the accuracy for diffusion model. Theoretical analysis shows that the convergence speed of diffusion model is determined by the spectral gap of the L-generator, which is the first non-zero eigenvalue. Experimental results demonstrate the theoretical results and the effectiveness of the proposed method.
1. This paper leverages the novel Chen’s estimation theory to diffusion model and provides a rigorous characteristic of the convergence speed using the spectral gap of the L-generator.
2. The theoretical analysis also innovates a novel algorithm using the L-generator. By choosing appropriate $a(x)$ and $b(x)$, we can balance the sampling speed and generation quality in diffusion model.
3. Numerical results verify the effectiveness of the proposed L-generator and provided several insights on the learning behavior under different sets of $a(x)$ and $b(x)$.
1.Although the paper proposes an interesting, principled mechanism for balancing sampling speed and sample quality in diffusion models, it does not provide clear tuning guidelines for $a(x)$ and $b(x)$. The guiding principles from Line 249 to 255 are still too general and lack concrete “go-to” defaults that reliably improve performance. Since practitioners need to choose both the functional forms $a(x)$ and $b(x)$, and the internal parameters in them, the resulting hyperparameter search can be extensive and may offset the performance gains. Providing a small set of recommended forms and default settings would make the method far more accessible.
2. Another limitation is the evaluation scope. The method has been tested with DDPM and ODE samplers, but not with more efficient samplers such as DDIM. It would be beneficial to see whether the technique also improves fast samplers, which further clarifies its impact on sampling efficiency.
1. Finding the point that balances the quality and efficiency might be time-consuming. Any guidance for finding such point?
2. I did not understand the relations of equation 5 and $\lambda_1$. Could you explain more on why equation 5 inspires the study of the principle eigenvalue?
3. To compute $\lambda_n$, what discretization procedure do you use, and how does the error scale with the discretization accuracy? And how large $n$ should we choose in practice? |
Fully human-written |
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EGEA-DM: Eigenvalue-Guided Explainable and Accelerated Diffusion Model |
Soundness: 2: fair
Presentation: 2: fair
Contribution: 2: fair
Rating: 4: marginally below the acceptance threshold
Confidence: 2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked. |
This paper aims to enhance the training efficiency, interpretability, and controllability of diffusion models. Drawing on ergodic theory, the authors proposed an Eigenvalue-Guided Explainable and Accelerated Diffusion Model (EGEA-DM). This approach leverages the principal eigenvalue of the L-generator to precisely regulate the forward diffusion speed, thereby enabling adaptive adjustment of reverse steps during both training and sampling phases.
1、Reinterpreting diffusion models through the lens of ergodic theory, while establishing a connection between the convergence rate of the forward process and the principal eigenvalue of the L-generator, is an interesting idea.
2、By tuning the coefficients of the L-generator according to its principal eigenvalue, the proposed method introduces a flexible mechanism that enables control over both the speed and stability of the diffusion process.
1、It appears that as the eigenvalue increases, the FID of the model's generated results may deteriorate. Given that the appropriate eigenvalue range is critical for balancing efficiency and generative performance, how should this range be determined? Is it necessary to conduct careful parameter tuning for different models individually?
2、Both qualitatively and quantitatively, the paper lacks comparisons between the proposed method, a baseline (i.e., the model without using the proposed method), and other SOTA approaches. Consequently, the true performance gains brought by the proposed method cannot be accurately evaluated. Furthermore, experiments are only conducted on two small-scale or low-resolution datasets—CIFAR-10 and CelebA-HQ-64, with no experimental validation of generalization on larger-scale or higher-resolution datasets.
3、The latest methods discussed in the "Related Work" section only date up to 2023, lacking discussions on the connections to and distinctions from more recent advances.
Please see weaknesses above. |
Lightly AI-edited |