Inference for Similarity and Alignability between Noisy High-Dimensional Datasets

The rapid growth of high-dimensional datasets across a wide range of scientific domains has created an urgent need for new statistical methods to compare distributions with underlying low-dimensional structure. Assessing similarity between high-dimensional datasets whose observations concentrate near low-dimensional manifolds is particularly challenging due to the nontrivial effects of noise in high dimensions. We propose a principled framework for statistical inference on the similarity and alignability of high-dimensional datasets with low-dimensional smooth signals under heterogeneous noise. The key idea is to link the spectral properties of the observed data matrices to the geometry of their underlying signal distributions. Under a manifold signal-plus-noise model, we build on the principal variances associated to the underlying signals and develop a scale- and rotation-invariant dissimilarity measure between two datasets that may differ in sample size and noise structure. We further construct an estimator of the dissimilarity and a statistical test of dataset alignability, namely, whether the dissimilarity vanishes. The proposed methodology and its theoretical guarantees under high-dimensional asymptotic settings draw on recent advances in random matrix theory (RMT). The proposed framework accommodates heterogeneous noise across datasets and provides a fast, theoretically grounded approach to comparing high-dimensional datasets with low-dimensional structures. Through extensive simulations and analyses of multiple single-cell datasets, we demonstrate that the proposed method substantially outperforms existing approaches.