Curvature of high-dimensional data (2511.02873v1)

Abstract: We consider the problem of estimating curvature where the data can be viewed as a noisy sample from an underlying manifold. For manifolds of dimension greater than one there are multiple definitions of local curvature, each suggesting a different estimation process for a given data set. Recently, there has been progress in proving that estimates of ``local point cloud curvature" converge to the related smooth notion of local curvature as the density of the point cloud approaches infinity. Herein we investigate practical limitations of such convergence theorems and discuss the significant impact of bias in such estimates as reported in recent literature. We provide theoretical arguments for the fact that bias increases drastically in higher dimensions, so much so that in high dimensions, the probability that a naive curvature estimate lies in a small interval near the true curvature could be near zero. We present a probabilistic framework that enables the construction of more accurate estimators of curvature for arbitrary noise models. The efficacy of our technique is supported with experiments on spheres of dimension as large as twelve.

• The paper presents a probabilistic framework that decodes true curvature from high-dimensional data using vMF noise models.
• It derives closed-form pushforward distributions of curvature estimates and shows significant bias in naive PCA-based approaches.
• Extensive experiments on high-dimensional spheres validate that the MLE-based curvature estimator remains robust even under substantial noise.

Curvature Estimation in High-Dimensional Data: A Probabilistic Framework

The paper "Curvature of high-dimensional data" (2511.02873) presents a rigorous analysis of curvature estimation for high-dimensional data clouds under the manifold hypothesis, with a focus on the practical and theoretical limitations of existing methods. The authors introduce a probabilistic framework for decoding curvature from noisy samples, derive closed-form expressions for curvature distributions under von Mises-Fisher (vMF) noise models, and demonstrate the efficacy of their approach through extensive experiments on high-dimensional spheres.

Motivation and Problem Formulation

Curvature estimation is central to data geometry, especially in the context of manifold learning where high-dimensional data is assumed to be sampled from a low-dimensional manifold embedded in $\mathbb{R}^n$. Traditional approaches estimate curvature by comparing tangent spaces at nearby points, typically using local PCA. However, recent literature reports systematic bias in such estimates, which increases with both ambient dimension and noise level. The paper provides theoretical arguments and empirical evidence that, in high dimensions, naive curvature estimates are highly unreliable and often fail to represent the true curvature.

Absolute Variation Curvature and Tangent Space Estimation

The authors focus on absolute variation curvature ($\omega$), an extrinsic measure quantifying the average change in tangent space orientation near a point. For a manifold $\mathcal{M} \subset \mathbb{R}^n$ of dimension $m$, the discrete variation curvature at $x$ in the direction of $y$ is defined as:

$$\Omega_y(x) = \frac{2 \sin(\frac{\theta(x, y)}{2})}{\|x - y\|_{\mathbb{R}^n}}$$

where $\theta(x, y)$ is the angle between tangent spaces at $x$ and $y$. In codimension one, this reduces to a function of the angle between normal vectors.

Figure 1: Tangent spaces of two nearby points on a circle, illustrating the geometric basis for curvature estimation via tangent space comparison.

Probabilistic Decoding: Pushforward of Noise Models

The core contribution is a probabilistic framework that models tangent space estimation as a random variable on the Grassmannian, with noise characterized by a vMF distribution. The curvature estimator $\Omega$ is then treated as a random variable whose distribution depends on both the noise parameters and the true curvature. The authors derive explicit pushforward formulas for the distribution of estimated curvature under vMF noise, enabling maximum likelihood estimation (MLE) of the true curvature $\omega$ from observed samples.

Figure 2: Histograms of curvature estimates for spheres $S^3$, $S^5$, and $S^{10}$, showing increasing bias in naive curvature estimation with dimension.

Figure 3: Theoretical probability density of naive curvature for spheres of dimension $3,5,10,20,50$, confirming the empirical bias observed in Figure 2.

Mixture Models and Generalization

Recognizing that real-world noise may not be well-approximated by a single vMF, the framework is extended to mixtures of vMF distributions. The pushforward operation is linear, so the curvature distribution under a mixture model is a weighted sum of the individual pushforwards. This universality allows the method to accommodate arbitrary noise models, provided they can be approximated by finite vMF mixtures.

Numerical Experiments and Performance Analysis

Extensive experiments are conducted on spheres $S^m_r$ of dimensions up to $m=12$ and radii $r=1,2$, with both perfect and noisy sampling. The naive curvature estimator exhibits strong bias, especially as $m$ or noise level $\sigma$ increases. In contrast, the decoded curvature estimator, based on the probabilistic pushforward and MLE, remains sharply peaked near the true value, even in high dimensions and under substantial noise.

Figure 4: Decoded curvature versus naive curvature with noisy sampling for $r=1$, $m=3,5$, and $\sigma = 0.01, 0.02, 0.05$. Decoded estimates are robust to noise and dimension, while naive estimates degrade rapidly.

Algorithmic Implementation

The practical implementation involves:

1. Tangent Space Estimation: Local PCA in a neighborhood of each point, with hyperparameters for neighborhood radius and local dimension.
2. Noise Characterization: Empirical or bootstrapped estimation of the tangent space error distribution, fitted to a vMF mixture.
3. Curvature Decoding: MLE of the true curvature parameter $\omega$ using the derived pushforward distribution and observed curvature samples.

The computational cost scales with the number of samples and the complexity of the mixture model. For high-dimensional spheres, the authors report using up to $10^8$ samples for robust estimation.

Theoretical and Practical Implications

The paper demonstrates that bias in naive curvature estimation is intrinsic to the method and exacerbated by high dimension and noise. The probabilistic decoding framework provides a principled remedy, yielding accurate curvature estimates under arbitrary noise models. The approach is theoretically generalizable to other curvature measures (e.g., mean, Gaussian, Ricci) and to manifolds of arbitrary codimension, though explicit pushforward formulas for Grassmannian-valued random variables remain an open problem.

Future Directions

Potential extensions include:

• Generalization to arbitrary codimension via matrix-valued vMF distributions on the Grassmannian.
• Application to real-world datasets (e.g., image patches, single-cell RNA-seq) with complex noise models.
• Integration with intrinsic curvature estimation methods and diffusion-based approaches.
• Development of scalable algorithms for large-scale, high-dimensional data.

Conclusion

This work provides a rigorous probabilistic framework for curvature estimation in high-dimensional data, addressing the limitations of naive tangent space methods. By modeling the noise and deriving explicit pushforward distributions, the authors achieve robust, unbiased curvature estimates even in challenging regimes. The methodology is broadly applicable and sets the stage for further advances in geometric data analysis and manifold learning.