Riemannian Simultaneous Inference for Tangent Vector Field Regression

We develop inference for manifold-valued population minimizers when each observation belongs to a different participant and only locally private messages reach the analyst. The method releases randomized tangent gradients and combines them through Riemannian stochastic approximation and Polyak-Ruppert averaging. Directly inserting a private data surrogate into a nonlinear loss can shift its population target, whereas conditional centring of the released gradient preserves the first-order equation. We introduce symmetric-pair regression (SPR) to estimate the asymptotic variance from the same private messages used for point estimation, without holding out participants or requesting a second release. We prove the central limit theorem and consistency of the fully transcript-based sandwich covariance and intrinsic Wald region under local differential privacy. Simulations across various statistical problems and manifolds support the predicted decrease in estimation error and near-nominal coverage under moderate privacy. An application to NHANES anthropometric data illustrates private estimation of a leading body-size direction and its uncertainty.

In this paper, we investigate the supremum-norm generalization error and the uniform inference for a specific class of kernel regression methods, namely the kernel gradient flows. Under the widely adopted capacity-source condition framework in the kernel regression literature, we first establish convergence rates for the supremum norm generalization error of both continuous and discrete kernel gradient flows under the source condition $s>α_0$, where $α_0\in(0,1)$ denotes the embedding index of the kernel function. Moreover, we show that these rates match the minimax optimal rates. Building on this result, we then construct simultaneous confidence bands for both continuous and discrete kernel gradient flows. Notably, the widths of the proposed confidence bands are also optimal, in the sense that their shrinkage rates are greater than, while can be arbitrarily close to, the minimax optimal rates.

Estimating the tangent spaces of a data manifold is a fundamental problem in geometric data analysis. The standard approach, Local Principal Component Analysis (LPCA), struggles in high-noise setting due to a critical trade-off in choosing the neighborhood size. Selecting an optimal size requires prior knowledge of the geometric and noise characteristics of the data that are often unavailable. In this paper, we propose a spectral method, Laplacian Eigenvector Gradient Orthogonalization (LEGO), that utilizes the global structure of the data to guide local tangent space estimation. Instead of relying solely on local neighborhoods, LEGO estimates the tangent space at each data point by orthogonalizing the gradients of low-frequency eigenvectors of the graph Laplacian. We provide two theoretical justifications of our method. First, a differential geometric analysis on the tubular neighborhood of a manifold shows that gradients of the low-frequency Neumann eigenfunctions of the tube align closely with the manifold's tangent bundle, while an eigenfunction with high gradient in directions orthogonal to the manifold lie deeper in the spectrum. Second, a random matrix theoretic analysis also demonstrates that low-frequency eigenvectors are robust to sub-Gaussian noise. These results allow us to derive the asymptotic scaling and stability of the estimated eigenvector gradients. Numerical experiments demonstrate that LEGO yields tangent space estimates that are significantly more robust to noise than those from LPCA, resulting in marked improvements in downstream tasks such as manifold learning, boundary detection, and local intrinsic dimension estimation.