Information-Theoretic Proofs for Diffusion Sampling (2502.02305v2)

Abstract: This paper provides an elementary, self-contained analysis of diffusion-based sampling methods for generative modeling. In contrast to existing approaches that rely on continuous-time processes and then discretize, our treatment works directly with discrete-time stochastic processes and yields precise non-asymptotic convergence guarantees under broad assumptions. The key insight is to couple the sampling process of interest with an idealized comparison process that has an explicit Gaussian-convolution structure. We then leverage simple identities from information theory, including the I-MMSE relationship, to bound the discrepancy (in terms of the Kullback-Leibler divergence) between these two discrete-time processes. In particular, we show that, if the diffusion step sizes are chosen sufficiently small and one can approximate certain conditional mean estimators well, then the sampling distribution is provably close to the target distribution. Our results also provide a transparent view on how to accelerate convergence by using additional randomness in each step to match higher-order moments in the comparison process.

• The paper provides novel information-theoretic proofs for discrete-time diffusion sampling, offering non-asymptotic convergence guarantees.
• It analyzes the diffusion sampling process by comparing it to a known comparison process and uses information-theoretic tools to bound the convergence divergence.
• The analysis, primarily for continuous data, has potential relevance for lossless text compression but requires adapting the methods for discrete data.

This paper provides a novel, discrete-time analysis of diffusion-based sampling methods for generative modeling, offering non-asymptotic convergence guarantees. Here's a breakdown of its contributions and relevance to lossless text data compression:

Main Contributions:

1. Discrete-Time Analysis: Unlike prior work that relies on continuous-time stochastic processes and then discretizes, this paper directly analyzes discrete-time stochastic processes. This approach simplifies the analysis and provides precise, non-asymptotic bounds on the convergence of the sampling distribution to the target distribution.
2. Comparison Process: The key idea is to introduce a "comparison process" ({Yk}) alongside the actual sampling process ({Zk}). The comparison process has a known Gaussian-convolution structure, making it easy to analyze. Crucially, the comparison process starts with a sample from the target distribution. The actual process that generates samples from an approximation of the target distribution, begins with Gaussian noise and attempts to map this to the target distribution.
3. Information-Theoretic Bounds: The paper leverages information-theoretic tools, specifically the Kullback-Leibler (KL) divergence and the I-MMSE relationship, to bound the discrepancy between the comparison process and the sampling process. The I-MMSE relationship connects mutual information and minimum mean-squared error in Gaussian noise channels.
4. Theorem 1 (Divergence Bound): The core result (Theorem 1) bounds the KL divergence between the joint distributions of the two processes. This bound depends on:
   • The covariance of the target distribution.
   • The step sizes used in the diffusion process.
   • How well the functions used in the sampling process approximate the conditional mean estimators of the comparison process.
5. Dimension-Free Bounds: The results are "dimension-free," meaning they don't explicitly depend on the dimensionality of the data. This suggests the potential for application to high-dimensional spaces.
6. Accelerated Convergence: Theorem~\ref{thm:moments} demonstrates that if one designs a sampling process that matches higher-order moments of the conditional distributions in the comparison process, the rate of convergence becomes O(n^-m), where m is related to the number of matched moments. Matching the mean is optimal with respect to relative entropy, but matching additional moments leads to faster convergence.

Applicability to Lossless Text Data Compression:

While the paper focuses on sampling from continuous distributions (e.g., images), the underlying principles have potential relevance to lossless text compression. Here's how the concepts might be leveraged and where limitations arise:

1. Entropy Limits: The paper's focus on approaching the target distribution in terms of KL divergence is directly related to the concept of entropy in data compression. The KL divergence measures the "extra bits" needed when using a model distribution (the sampling process) instead of the true distribution (the target). Minimizing KL divergence is analogous to approaching the entropy limit, which is the theoretical minimum number of bits per symbol achievable for a given source.
2. Redundancy Reduction: Diffusion models, as analyzed in the paper, aim to model the complex dependencies within a data distribution. For text data, this translates to capturing the statistical relationships between characters, words, and phrases. By accurately modeling these dependencies, a diffusion-based approach could, in principle, lead to better redundancy reduction compared to simpler models. The use of conditional means relates to prediction, where, by predicting the next item in a sequence, the prediction can be encoded with fewer bits.
3. Algorithmic Efficiency: A significant challenge in lossless compression is the trade-off between compression ratio and computational efficiency. Traditional methods like arithmetic coding are relatively fast, while neural network-based approaches often have high computational costs. The paper's analysis of step sizes and convergence rates provides insights into this trade-off. Smaller step sizes improve accuracy (lower KL divergence, better compression) but require more computation.
4. Discrete vs Continuous: The approach taken in the paper needs to be adapted for lossless text compression, where the underlying distribution is discrete, rather than continuous, as assumed in the paper. This is a major point to be addressed for applicability to text compression.
5. Beyond Entropy Limits? Standard information theory dictates that one cannot compress below the entropy limit for a given model. However, the paper's focus on increasingly accurate approximations of conditional distributions, especially the potential for superlinear convergence with higher-order moment matching (Theorem 4), raises a question. Could a diffusion-based model, by learning extremely complex dependencies, practically achieve compression ratios that appear to surpass the entropy limits calculated based on simpler models (e.g., n-gram models)? The theoretical limit remains, but the practical limit, with a sufficiently powerful model, might be tighter.

Comparison with Arithmetic Coding:

• Arithmetic Coding: A highly efficient, widely-used entropy coding technique. It represents data as a single fraction, achieving compression very close to the entropy limit for a given probabilistic model (e.g., a Markov model).
• Diffusion Models (Potential): Could potentially capture more complex, long-range dependencies in text than traditional models used with arithmetic coding. This could lead to better compression if the computational cost can be managed. The paper's analysis suggests a path towards understanding and controlling this trade-off.
• Computational Complexity: This is the key difference. Arithmetic coding with a well-chosen model is generally fast. Diffusion models, especially those achieving high accuracy, are likely to be much slower.

Potential Improvements, Limitations, and Future Research Directions (Specifically for Text Compression):

1. Discrete Distributions: The most crucial extension is adapting the analysis to discrete distributions, which are fundamental to text data. This would involve replacing Gaussian noise with appropriate discrete noise models and reformulating the comparison process.
2. Computational Efficiency: Practical text compression requires very fast algorithms. Research is needed to develop diffusion-based compression methods that are computationally competitive with existing techniques. This may include:
   • Exploring efficient approximations of the conditional mean estimators.
   • Optimizing the step sizes and number of steps for the best compression/speed trade-off.
   • Leveraging specialized hardware (GPUs, TPUs) for acceleration.
   • Considering the use of the sampling process of Equation 15.
3. Model Complexity: The paper highlights the importance of accurate conditional mean estimation. For text, this means building models that can effectively capture long-range dependencies and contextual information. Transformers and LLMs are potential candidates, but their computational cost needs careful consideration.
4. Adaptive Models: Effective text compression often relies on adaptive models that update their statistics as they process the data. Integrating adaptivity into a diffusion-based compression framework would be a valuable research direction.
5. Hybrid Approaches: Combining diffusion models with existing compression techniques (e.g., using a diffusion model to generate probabilities for arithmetic coding) could offer a pragmatic way to improve compression ratios while maintaining reasonable computational efficiency.

In summary, while this paper presents a theoretical analysis of diffusion models in a continuous setting, its core ideas—approaching a target distribution via a carefully constructed stochastic process, bounding divergence using information-theoretic tools, and understanding the trade-offs between accuracy and computational cost—are highly relevant to the broader goal of improving lossless text data compression. Bridging the gap between the continuous theory and the discrete nature of text, and addressing the computational challenges, are key areas for future research.