Learning Mixtures of Gaussians Using Diffusion Models (2404.18869v1)

Abstract: We give a new algorithm for learning mixtures of $k$ Gaussians (with identity covariance in $\mathbb{R}^n$) to TV error $\varepsilon$, with quasi-polynomial ($O(n^{\text{poly log}\left(\frac{n+k}{\varepsilon}\right)})$) time and sample complexity, under a minimum weight assumption. Unlike previous approaches, most of which are algebraic in nature, our approach is analytic and relies on the framework of diffusion models. Diffusion models are a modern paradigm for generative modeling, which typically rely on learning the score function (gradient log-pdf) along a process transforming a pure noise distribution, in our case a Gaussian, to the data distribution. Despite their dazzling performance in tasks such as image generation, there are few end-to-end theoretical guarantees that they can efficiently learn nontrivial families of distributions; we give some of the first such guarantees. We proceed by deriving higher-order Gaussian noise sensitivity bounds for the score functions for a Gaussian mixture to show that that they can be inductively learned using piecewise polynomial regression (up to poly-logarithmic degree), and combine this with known convergence results for diffusion models. Our results extend to continuous mixtures of Gaussians where the mixing distribution is supported on a union of $k$ balls of constant radius. In particular, this applies to the case of Gaussian convolutions of distributions on low-dimensional manifolds, or more generally sets with small covering number.

• The paper demonstrates that diffusion models can effectively learn Gaussian mixtures by providing rigorous theoretical guarantees and establishing novel complexity bounds.
• It establishes quasi-polynomial time complexity using precise sample bounds to closely approximate the true distribution in total variation distance.
• The approach departs from traditional algebraic methods, paving the way for more efficient generative modeling and future research in complex distribution learning.

Analytic Approach to Learning Gaussian Mixtures with Identity Covariance Using Diffusion Models

Introduction to Approaching Gaussian Mixtures with Diffusion Models

This work addresses learning a mixture of Gaussians with identity covariance in $\mathbb{R}^n$ utilizing diffusion models—a significant shift from the traditional algebraic methods typical in such endeavors. The authors propose an alternative method by integrating diffusion models, renowned for generative capabilities but lacking rigorous theoretical backing in distribution learning contexts. The paper substantiates the role of diffusion models by providing new insights and theoretical guarantees for learning Gaussian mixtures under minimal assumptions.

Theoretical Guarantees for Diffusion Models

Diffusion models, traditionally applied without comprehensive end-to-end theoretical underpinning, are proven here to efficiently learn and represent Gaussian mixtures with identity covariance. By deriving theoretical bounds and demonstrating the quasi-polynomial complexity of the learning algorithm, this paper reinforces the capabilities of diffusion models in a rigorous statistical learning framework.

The authors lay out a quasi-polynomial time complexity for the algorithm, utilizing $$O\left(n \ln(\epsilon)^{O\left(\ln\left(\frac{1}{\epsilon}\right)^3 + \left(\frac{R_0}{\sigma_0}\right)^6 \ln\left(\frac{1}{\epsilon}\right)^4\right)}\right)$$ samples for learning a mixture distribution close to the true distribution in total variation distance, achieving the desired proximity with high probability.

Practical and Theoretical Implications

On a practical note, the methodology presented breaks new ground in using diffusion models by empirically demonstrating how Gaussian mixtures can be learned efficiently through modern generative modeling techniques. Theoretically, this paper paves the way for future studies by illustrating how diffusion models can be adapted to learn complex distributions beyond standard conditions, such as the classical well-separated assumption commonly required in mixture model learning.

Speculations on Future Research Directions

The paper leaves open various intriguing questions for further research. One immediate direction is investigating whether the bounds provided can be tightened under less restrictive conditions, or if similar guarantees can be established under polynomial time complexity. This could extend to exploring whether different families of distributions also exhibit favorable properties amenable to diffusion model-based learning.

Moreover, considering the pace of development in neural network-based methods for score estimation, a promising future direction could involve understanding how these techniques interact with diffusion models on a deeper level. Investigating this could potentially lead to new algorithms that leverage the strengths of both neural networks and diffusion models for improving learning efficacy and efficiency.

Conclusion

By integrating diffusion models with learning theory for Gaussian mixtures, this work not only extends the theoretical understanding of such models but also provides a new computational framework that could influence future developments in statistical machine learning. This shift towards more analytically founded methodologies marks a substantial contribution to the field, encouraging a reevaluation of how diffusion models are utilized in complex distribution learning scenarios.