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Your diffusion model secretly knows the dimension of the data manifold (2212.12611v5)

Published 23 Dec 2022 in cs.LG and stat.ML

Abstract: In this work, we propose a novel framework for estimating the dimension of the data manifold using a trained diffusion model. A diffusion model approximates the score function i.e. the gradient of the log density of a noise-corrupted version of the target distribution for varying levels of corruption. We prove that, if the data concentrates around a manifold embedded in the high-dimensional ambient space, then as the level of corruption decreases, the score function points towards the manifold, as this direction becomes the direction of maximal likelihood increase. Therefore, for small levels of corruption, the diffusion model provides us with access to an approximation of the normal bundle of the data manifold. This allows us to estimate the dimension of the tangent space, thus, the intrinsic dimension of the data manifold. To the best of our knowledge, our method is the first estimator of the data manifold dimension based on diffusion models and it outperforms well established statistical estimators in controlled experiments on both Euclidean and image data.

Citations (17)

Summary

  • The paper introduces a novel method that leverages diffusion models to estimate the intrinsic dimension by analyzing score function orthogonality near the data manifold.
  • It employs singular value decomposition of score vectors at low noise levels to accurately determine manifold dimensions, outperforming traditional techniques on synthetic and MNIST datasets.
  • The study reveals that diffusion models implicitly capture underlying data structure, offering robust insights for advancing manifold learning and generative modeling applications.

Summary of "Your diffusion model secretly knows the dimension of the data manifold"

The paper "Your diffusion model secretly knows the dimension of the data manifold" presents a novel methodology for estimating the intrinsic dimension of data manifolds through the use of diffusion models. The authors assert that despite diffusion models being non-explicitly tailored to account for the intrinsic dimension of data, they inherently encapsulate enough information to make this estimation.

Diffusion models operate by approximating the score function, which is the gradient of the log density of a data distribution corrupted by noise. The authors demonstrate that, as the noise level diminishes, the score function aligns itself orthogonally towards the data manifold. This orthogonality signifies the direction of maximum likelihood increase, thereby granting access to an approximation of the normal bundle of the manifold. Consequently, the tangent space, and thus the intrinsic dimension of the data manifold, can be estimated.

Key Insights and Methodology

The proposed method stands out as the first to leverage diffusion models for estimating the data manifold's dimension. The approach involves:

  • Training a diffusion model to approximate the score function across varying noise levels.
  • Utilizing small noise levels to approximate the normal space of the manifold.
  • Computing the singular value decomposition (SVD) of the score vectors, where the number of near-zero singular values corresponds to the manifold's dimension.

The paper provides theoretical support, including proofs that verify the approach, showing that the score at any point close to the data manifold aligns with the orthogonal projection on the manifold. This alignment is crucial for determining the dimensionality.

Experimental Evaluation

The method was empirically tested on various datasets, including both synthetic Euclidean and image data, as well as the MNIST dataset. On well-constructed synthetic manifolds, including kk-spheres and complex image manifolds, the proposed approach consistently achieved accurate dimension estimates, often outperforming traditional methods such as Maximum Likelihood Estimation (MLE) and Local PCA.

On the MNIST dataset, for which the intrinsic dimension is not predetermined, the method's estimates were validated by comparing with auto-encoder reconstruction errors across varying latent dimensions. The results suggest higher dimensions than some previous estimates, indicating a potentially deeper understanding of the dataset's structure.

Theoretical and Practical Implications

This research contributes to the theoretical understanding of diffusion models, showing their implicit capability to reveal data characteristics beyond their primary generative purpose. Practically, the method provides a robust tool for modeling complex, high-dimensional datasets without requiring pre-defined dimensional information, potentially benefiting various applications that deal with manifold-structured data.

Future Directions

The paper opens avenues for further exploration into diffusion models, suggesting the possibility of extending their use to other types of data and exploring their capabilities in manifold learning. Moreover, enhancing the robustness and scalability of the method in diverse and complex data scenarios could yield further practical applications and theoretical insights into generative modeling.

Overall, the paper introduces a compelling and insightful perspective to manifold dimension estimation through diffusion models, positioning itself as a significant contribution to the field of deep generative models and manifold learning.