Manifold Diffusion Geometry: Curvature, Tangent Spaces, and Dimension (2411.04100v2)

Abstract: We introduce novel estimators for computing the curvature, tangent spaces, and dimension of data from manifolds, using tools from diffusion geometry. Although classical Riemannian geometry is a rich source of inspiration for geometric data analysis and machine learning, it has historically been hard to implement these methods in a way that performs well statistically. Diffusion geometry lets us develop Riemannian geometry methods that are accurate and, crucially, also extremely robust to noise and low-density data. The methods we introduce here are comparable to the existing state-of-the-art on ideal dense, noise-free data, but significantly outperform them in the presence of noise or sparsity. In particular, our dimension estimate improves on the existing methods on a challenging benchmark test when even a small amount of noise is added. Our tangent space and scalar curvature estimates do not require parameter selection and substantially improve on existing techniques.

• The paper introduces novel diffusion geometry estimators that robustly compute curvature, tangent spaces, and dimensionality on manifold data despite noise.
• It replaces hard neighborhood methods with a continuous, kernel-driven approach, eliminating the need for parameter selection and enhancing accuracy.
• Numerical experiments on 12 diverse manifolds demonstrate that these techniques significantly outperform classical Riemannian methods in noisy conditions.

Manifold Diffusion Geometry: Curvature, Tangent Spaces, and Dimension

The paper "Manifold Diffusion Geometry: Curvature, Tangent Spaces, and Dimension," by Iolo Jones, addresses a crucial gap in geometric data analysis by advancing computational methods using diffusion geometry to estimate curvature, tangent spaces, and dimensions of manifold data. Leveraging diffusion geometry allows the formulation of novel estimators that surpass traditional Riemannian geometry methods in robustness, especially in the presence of noise and sparsity.

Advances in Diffusion Geometry

The paper begins by situating itself within the manifold hypothesis, which assumes that data lies on a manifold, making Riemannian geometry applicable. However, real-world data often pose challenges like discreteness and noise, impairing the direct use of Riemannian geometry. By employing diffusion geometry—which utilizes the heat flow on manifolds—a robust framework is established for estimating manifold properties with high statistical performance even amidst significant data imperfections.

Core Contributions

• Curvature Estimation: The paper introduces estimators for Riemannian curvature, Ricci curvature, and particularly the scalar curvature, which traditionally receive limited attention in computational contexts. These are formulated via the Laplace operator tied to diffusion processes, yielding enhanced robustness across noisy datasets.
• Tangent Space and Dimension Estimation: New estimators are developed for calculating tangent spaces and the dimension of manifolds. These estimators eliminate the typical parameter selection present in previous methods, leading to significant improvements in accuracy under conditions of noise and data sparsity.

Numerical Results

The empirical evaluations demonstrate the methods' efficacy, especially the dimension estimation technique, which performs comparably on pristine datasets and surpasses others amid noise. By testing 12 diverse manifolds with various densities and noise levels, diffusion geometry consistently exhibits superior robustness, reflected in concrete numerical benchmarks.

Methodological Stance

The paper presents a clear departure from the traditional "hard neighbourhood" paradigm that relies on discrete point neighborhoods for estimating manifold properties. Instead, it employs a continuous "soft neighborhood," leveraging kernel methods to enhance robustness without sacrificing accuracy.

Implications and Future Directions

Practically, these improvements suggest significant potential for diffusion geometry in machine learning and data-driven sciences, where reliability in low-quality data conditions is often required. Theoretically, the work invites further exploration of manifold-based geometry in non-manifold contexts, perhaps extending into general probabilistic spaces.

The discussion on robustness linked to intrinsic low-dimensionality and the inherent challenges associated with higher-dimensional settings provides insight into the limitations and scalability concerns of the proposed methods. As AI and ML continue to evolve, the paper implicitly calls for embedding these geometric insights into more generalized learning frameworks, possibly inspiring novel architectures or regularization techniques.

In conclusion, by addressing robustness and parameterization challenges, this research contributes substantially to computational methods in geometric data analysis, with promising pathways for application in noisy real-world scenarios and complex data landscapes. Future research should consider the analysis of curvature as feature inputs for machine learning algorithms, potentially enhancing their performance on higher-dimensional, noise-prone data.