# Inference for Similarity and Alignability between Noisy High-Dimensional Datasets

arXiv: https://arxiv.org/abs/2511.21074 · HTML: https://mlsanity.com/paper/2511.21074/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 58 references that are in the map.
- First posted on arXiv 26 Nov 2025.

## Builds on (most cited first, 14 of 14)

- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Unsupervised Domain Adaptation via Discriminative Manifold Propagation](https://mlsanity.com/paper/2008.10030/) (2020)
- [A new graph-based two-sample test for multivariate and object data](https://mlsanity.com/paper/1307.6294/) (2013)
- [Inference for Heteroskedastic PCA with Missing Data](https://mlsanity.com/paper/2107.12365/) (2021)
- [On the Optimality of Kernel-Embedding Based Goodness-of-Fit Tests](https://mlsanity.com/paper/1709.08148/) (2017)
- [ScreeNOT: Exact MSE-Optimal Singular Value Thresholding in Correlated Noise](https://mlsanity.com/paper/2009.12297/) (2020)
- [Gaussian Graphical Model Estimation with False Discovery Rate Control](https://mlsanity.com/paper/1306.0976/) (2013)
- [Learning Low-Dimensional Nonlinear Structures from High-Dimensional Noisy Data: An Integral Operator Approach](https://mlsanity.com/paper/2203.00126/) (2022)
- [Matrix denoising for weighted loss functions and heterogeneous signals](https://mlsanity.com/paper/1902.09474/) (2019)
- [Is your data alignable? Principled and interpretable alignability testing and integration of single-cell data](https://mlsanity.com/paper/2308.01839/) (2023)
- [A weighted edge-count two-sample test for multivariate and object data](https://mlsanity.com/paper/1604.06515/) (2016)
- [How do kernel-based sensor fusion algorithms behave under high dimensional noise?](https://mlsanity.com/paper/2111.10940/) (2021)
- [Kernel spectral joint embeddings for high-dimensional noisy datasets using duo-landmark integral operators](https://mlsanity.com/paper/2405.12317/) (2024)
- [A universal test on spikes in a high-dimensional generalized spiked model and its applications](https://mlsanity.com/paper/2203.06924/) (2022)

## Built on by (most cited first, 1 of 1)

- [I-BBS: Coordinate-Free Inference of Latent Sub-Manifolds Using Random Distance Matrix Theory](https://mlsanity.com/paper/2606.29675/) (2026)

## Abstract (arXiv)

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.