When Diffusion Model Can Ignore Dimension: An Entropy-Based Theory

Abstract: Diffusion models perform remarkably well on high-dimensional data such as images, often using only a modest number of reverse-time steps. Despite this practical success, existing convergence theory does not fully explain why such samplers remain efficient in high dimensions. Many prior KL guarantees bound the discretization error in terms of the ambient dimension, while other improved results replace this dependence using intrinsic-dimensional or geometric structure assumptions. In this work, we develop an alternative information-theoretic perspective on diffusion sampler convergence. We prove that, for Gaussian mixture targets, the discretization error is controlled by the Shannon entropy of the latent mixture component rather than by the ambient dimension. Consequently, the leading step complexity scales linearly with latent entropy and depends only logarithmically on the second moment of the data. Our analysis also extends to discrete target distributions, where the relevant complexity is the entropy of the target rather than the dimension of the embedding space. These results suggest that diffusion sampling can remain efficient in high-dimensional spaces when the data distribution admits a compact latent representation, as is widely believed to be the case for natural images.

• The paper demonstrates that latent Shannon entropy, not ambient dimension, controls the discretization error in diffusion sampling for Gaussian mixture models.
• It employs the I-MMSE identity to convert discretization error into an MMSE area functional, yielding explicit non-asymptotic KL divergence bounds.
• Practical implications include efficient sampling in generative models through compact latent representations in applications like vision and language.

Entropy-Based Complexity Bounds in Diffusion Model Sampling

Motivation and Background

Diffusion models have achieved notable empirical efficacy in generating high-dimensional data (images, audio, etc.) while only employing a moderate number of reverse steps, defying theoretical expectations that complexity should scale inextricably with the ambient data dimension. Conventional convergence guarantees for diffusion samplers typically exhibit explicit ambient dimension dependence, or at best appeal to geometric notions like intrinsic dimension or covering numbers. However, neither ambient nor intrinsic dimension appears to fully explain the practical efficiency witnessed when sampling from high-dimensional data with compact latent structure.

The present work develops a fundamentally information-theoretic analysis of diffusion model sampling complexity. Rather than relying on the ambient or intrinsic dimension, or on geometric conditions on the data manifold, the paper identifies the Shannon entropy of the latent variable as the critical, governing complexity parameter for discretization error in the sampler—at least for data distributions that are (possibly infinite) Gaussian mixtures indexed by a discrete latent code.

Main Results: Entropy Controls Discretization, Not Dimension

The authors consider a model in which $Z = U + \epsilon G_0$, with $U = \mu_J$ for some discrete latent $J$, mixture centers $\{\mu_j\}$, and independent Gaussian noise $G_0$, so that $Z$ is a Gaussian mixture in $\mathbb{R}^d$. The dominant source of statistical uncertainty is the latent variable $J$ rather than the full ambient vector. The reverse SDE for diffusion sampling then becomes a denoising process for $J$ masked by the Gaussian channel induced by adding noise.

The principal theoretical contribution is a non-asymptotic upper bound on the KL divergence between the discretized sampler's output distribution and the true forward process (smoothed to avoid singularity at the final time). Critically, for a carefully constructed $K$-step discretization grid, the leading discretization error scales as

$U = \mu_J$0

where $U = \mu_J$1 is the Shannon entropy of the latent code, $U = \mu_J$2 is the second moment (energy) of the target distribution, and $U = \mu_J$3 is the inverse scale at the terminal SNR. The important point is that, apart from initialization and score-approximation terms, the ambient dimension $U = \mu_J$4 is absent from the leading term.

Moreover, this approach generalizes to discrete target distributions as $U = \mu_J$5, meaning that for data distributions over codebooks or tokens, the complexity is determined by the entropy of the target, not the size of the codebook or the dimension of the embedding.

The theoretical method hinges on converting the discretization error to an "MMSE (minimum mean-square error) area functional" via the I-MMSE identity, so that mutual information and entropy directly control the sampling error. This manipulation leverages the fact that mutual information for the discrete latent can be upper bounded by $U = \mu_J$6, and that the entropy bound applies even when the number of mixture components is countably infinite, provided $U = \mu_J$7.

Technical Novelty

A key element is the adaptation of the typical score-based reverse-time sampler to what the authors call a latent posterior freezing rule: on each discretization interval, the sampler freezes the posterior mean of the latent center (as opposed to the full data variable). The algorithm remains completely score-based (i.e., it requires only evaluations of the learned score function), but the analysis exposes and exploits the underlying information-theoretic structure.

The explicit construction of a hybrid $U = \mu_J$8-grid (regularized SNR schedule) furnishes a step-wise partition where the MMSE can be tightly controlled by $U = \mu_J$9 in the low SNR regime, and by the entropy $J$0 in the high SNR regime. This yields a sampling schedule that allocates steps most efficiently with respect to the actual complexity of the uncertainty, as measured by entropy—not by the number or structure of mixture components.

Implications

Theoretical Implications:

This work demonstrates rigorously that sample complexity of diffusion-model-based samplers can be independent of ambient dimension for a broad class of structured distributions, with step bounds instead governed by the entropy of the relevant latent space. This reframes discourse on sampling hardness, emphasizing structure and uncertainty over linear algebraic or geometric quantity.

The findings also provide a unified lens for understanding earlier results which depended on geometric or combinatorial characteristics—such as covering numbers, or the number of mixture components—highlighting that entropy provides a strictly tighter measure, especially pronounced when the latent distribution is highly non-uniform (since $J$1 in such cases).

The approach is distinguished from other dimension-free analyses (e.g., "Dimension-free Analysis of Diffusion-based Generative Models for Mixtures" [li2025dimension]) by its focus on KL divergence and entropy rather than total variation distance and cardinality- or geometry-based metrics. For highly structured or quantized data appearing in vision and language, this analysis is particularly relevant.

Practical Implications:

Many practical models (e.g., vector-quantized VAEs, VQGANs, or image tokenizers like TiTok) represent data using compressed discrete latent codes decoded into high-dimensional ambient space. The results here imply that, when such compact representations are achievable and the entropy is low, diffusion-based generators can efficiently sample convincing data with step counts that scale only with entropy. For instance, TiTok-L-32 encodes a $J$2 image as only 32 tokens from a codebook of size 4096, yielding an upper bound of $J$3 nats worst-case entropy—far below the ambient pixel dimension.

For language and discrete-sequence generation, the bounds here imply sharper vocabulary and length dependence than those given by $J$4-leaping analyses in discrete diffusion, since here only the entropy (not the product of dimension and vocabulary size) controls discretization error for Gaussianized embeddings.

Limitations and Outlook:

The analysis applies specifically to Gaussian mixtures and requires finite entropy and a finite second moment. Extensions to more general, unstructured distributions require nontrivial additional ideas; there is no guarantee that arbitrary high-dimensional data will always admit the representations considered here. Nevertheless, the paper suggests that for many practical data distributions encountered in generative modeling, entropy-based complexity measures may accurately predict empirical sampling efficiency.

Potential Future Directions

• Extension to non-Gaussian and non-mixture models: While this work is limited to Gaussian mixtures, it motivates seeking similar entropy-based control for broader families of distributions (e.g., heavy-tailed, multimodal, or data with intricate dependency structures).
• Latent entropy estimation and control in learned models: The link between compact latent representations (e.g., in VQVAEs) and sampling efficiency provides both a diagnostic metric and a target for representation learning.
• Adaptive, entropy-aware discretization schedules: Given practical model estimates of latent entropy, samplers could dynamically allocate computational budget according to uncertainty, potentially improving generation quality and speed.
• Score-approximation error analysis in the entropy regime: The present bound isolates the score-approximation contribution; future work could elucidate the conditions under which this term becomes the dominant bottleneck in practice.

Conclusion

This paper develops an entropy-based theory for understanding the efficiency of diffusion model samplers for high-dimensional, structured data distributions. By proving that discretization error can be controlled by the latent Shannon entropy rather than by ambient dimension, the work establishes an information-theoretic criterion for sampling complexity. This perspective both sharpens theoretical understanding of the practical efficiency of diffusion models and points toward more nuanced complexity measures and adaptive algorithms for generative modeling.