A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models (2406.03537v2)

Abstract: High-dimensional data commonly lies on low-dimensional submanifolds, and estimating the local intrinsic dimension (LID) of a datum -- i.e. the dimension of the submanifold it belongs to -- is a longstanding problem. LID can be understood as the number of local factors of variation: the more factors of variation a datum has, the more complex it tends to be. Estimating this quantity has proven useful in contexts ranging from generalization in neural networks to detection of out-of-distribution data, adversarial examples, and AI-generated text. The recent successes of deep generative models present an opportunity to leverage them for LID estimation, but current methods based on generative models produce inaccurate estimates, require more than a single pre-trained model, are computationally intensive, or do not exploit the best available deep generative models: diffusion models (DMs). In this work, we show that the Fokker-Planck equation associated with a DM can provide an LID estimator which addresses the aforementioned deficiencies. Our estimator, called FLIPD, is easy to implement and compatible with all popular DMs. Applying FLIPD to synthetic LID estimation benchmarks, we find that DMs implemented as fully-connected networks are highly effective LID estimators that outperform existing baselines. We also apply FLIPD to natural images where the true LID is unknown. Despite being sensitive to the choice of network architecture, FLIPD estimates remain a useful measure of relative complexity; compared to competing estimators, FLIPD exhibits a consistently higher correlation with image PNG compression rate and better aligns with qualitative assessments of complexity. Notably, FLIPD is orders of magnitude faster than other LID estimators, and the first to be tractable at the scale of Stable Diffusion.

• The paper introduces FLIPD as a novel estimator for local intrinsic dimension that leverages diffusion models to reduce computational load.
• It integrates the Fokker-Planck equation with a single diffusion model, providing unbiased LID estimates in linear subspace scenarios.
• Empirical results on synthetic and image datasets demonstrate FLIPD's robustness across various scales and dimensionalities.

Efficient Local Intrinsic Dimension Estimation with Diffusion Models

The paper "A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models" presents a novel methodology for estimating the local intrinsic dimension (LID) of data leveraging diffusion models (DMs). Understanding the intrinsic dimension of a dataset is critical, as it provides insights into its underlying complexity. This is particularly pertinent for high-dimensional datasets, often observed to lie on lower-dimensional manifolds according to the manifold hypothesis. The intrinsic dimension corresponds to the number of degrees of variability in the data, which can aid in tasks such as out-of-distribution (OOD) detection, generalization analysis in neural networks, adversarial sample identification, and AI-generated content discernment.

Key Contributions

1. FLIPD Estimator: This paper introduces the FLIPD (Fokker-Planck-based LID estimation) leveraging the Fokker-Planck equation associated with a DM. It circumvents the computational drawbacks of traditional methods and existing model-based approaches that often demand multiple DMs, are inaccurate, or computationally intensive.
2. Efficiency with a Single DM: The paper fundamentally improves on the LIDL estimator by integrating it with a single diffusion model instead of training numerous normalizing flows. The Fokker-Planck equation enables direct estimation of LID from the learned score function of the DM, providing significant computational advantages.
3. Theoretical Foundations: The researchers offer theoretical underpinnings for their estimator, particularly in contexts involving linear subspaces. They demonstrate that the partial derivative of the log-density, from a Gaussian convolution as the log standard deviation approaches negative infinity, produces an unbiased LID estimate.
4. Empirical Demonstration: The authors benchmark the efficacy of FLIPD on synthetic datasets with known intrinsic dimensions and against image datasets where complexity can be qualitatively assessed. FLIPD consistently provides robust LID estimates across various scales and dimensionalities.

Implications and Future Work

The ability to efficiently estimate local intrinsic dimensions has significant implications in both theoretical and practical domains. Theoretically, it enhances our understanding of data manifolds, particularly within the framework of deep generative models and their capacity to learn manifold structures. Practically, FLIPD could be instrumental in numerous AI applications such as model regularization, OOD detection, anomaly detection, and more robust AI-generated content assessment.

Future developments could focus on improving the integration of LID estimates with real-time applications, especially in scenarios where computational resources are constrained, or where models must dynamically adapt to shifting data distributions. Moreover, exploring extensions of FLIPD to handle even more complex representations could further its applicability.

In summary, this paper provides a substantial contribution to the field by simplifying and expediting a previously resource-intensive process. It opens new pathways for leveraging generative models for structural data analysis, enhancing our ability to interpret and utilize high-dimensional data across various AI domains.