# Selective Inference for Clustering Under Unknown Covariance

arXiv: https://arxiv.org/abs/2610.17458 · HTML: https://mlsanity.com/paper/2610.17458/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 66 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Builds on (most cited first, 30 of 34)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [C-Pack: Packed Resources For General Chinese Embeddings](https://mlsanity.com/paper/2309.07597/) (2023)
- [From Louvain to Leiden: guaranteeing well-connected communities](https://mlsanity.com/paper/1810.08473/) (2018)
- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Exact post-selection inference, with application to the lasso](https://mlsanity.com/paper/1311.6238/) (2013)
- [Optimal Inference After Model Selection](https://mlsanity.com/paper/1410.2597/) (2014)
- [Selective Inference for Hierarchical Clustering](https://mlsanity.com/paper/2012.02936/) (2020)
- [Testing for a Change in Mean After Changepoint Detection](https://mlsanity.com/paper/1910.04291/) (2019)
- [More Powerful Conditional Selective Inference for Generalized Lasso by Parametric Programming](https://mlsanity.com/paper/2105.04920/) (2021)
- [Selective inference for k-means clustering](https://mlsanity.com/paper/2203.15267/) (2022)
- [An Overview on the Estimation of Large Covariance and Precision Matrices](https://mlsanity.com/paper/1504.02995/) (2015)
- [Regularized EM Algorithms: A Unified Framework and Statistical Guarantees](https://mlsanity.com/paper/1511.08551/) (2015)
- [Splitting strategies for post-selection inference](https://mlsanity.com/paper/2102.02159/) (2021)
- [Influential Feature PCA for high dimensional clustering](https://mlsanity.com/paper/1407.5241/) (2014)
- [Data fission: splitting a single data point](https://mlsanity.com/paper/2112.11079/) (2021)
- [Data thinning for convolution-closed distributions](https://mlsanity.com/paper/2301.07276/) (2023)
- [Statistical Significance for Hierarchical Clustering](https://mlsanity.com/paper/1411.5259/) (2014)
- [Permutation methods for factor analysis and PCA](https://mlsanity.com/paper/1710.00479/) (2017)
- [Generalized Data Thinning Using Sufficient Statistics](https://mlsanity.com/paper/2303.12931/) (2023)
- [Finite mixture models do not reliably learn the number of components](https://mlsanity.com/paper/2007.04470/) (2020)
- [More powerful selective inference for the graph fused lasso](https://mlsanity.com/paper/2109.10451/) (2021)
- [Selective inference for clustering with unknown variance](https://mlsanity.com/paper/2301.12999/) (2023)
- [Inference after latent variable estimation for single-cell RNA sequencing data](https://mlsanity.com/paper/2207.00554/) (2022)
- [Decomposing Gaussians with Unknown Covariance](https://mlsanity.com/paper/2409.11497/) (2024)
- [Statistical significance of variables driving systematic variation](https://mlsanity.com/paper/1308.6013/) (2013)
- [Markov Chain Monte Carlo Significance Tests](https://mlsanity.com/paper/2310.04924/) (2023)
- [Statistical Testing Framework for Clustering Pipelines by Selective Inference](https://mlsanity.com/paper/2603.18413/) (2026)
- [Inference conditional on selection: a review](https://mlsanity.com/paper/2604.09779/) (2026)
- [Selective inference after convex clustering with ℓ1 penalization](https://mlsanity.com/paper/2309.01492/) (2023)
- [Post-clustering Inference under Dependence](https://mlsanity.com/paper/2310.11822/) (2023)

## Abstract (arXiv)

Unsupervised clustering is widely used to discover underlying group structure, after which researchers often seek to validate these groups via post-hoc hypothesis testing on the population means. However, using the same data both to form clusters and to test for a difference in means inflates the Type I error rate. Recent work in selective inference has proposed solutions when the underlying data are drawn from a multivariate Gaussian distribution with a known covariance matrix or an unknown isotropic covariance matrix. We propose a new selective test for the difference in means between two clusters that provably controls the Type I error rate under an arbitrary unknown covariance matrix. Our test statistic does not require estimating the feature covariance matrix; instead, it leverages a Monte Carlo sampling procedure over the set of orthogonal matrices. We show that this new selective test can be computed efficiently, and we derive closed-form solutions when clusters are formed using k-means or hierarchical clustering. In simulations, we demonstrate that our proposed p-value controls the selective Type I error rate and achieves substantial power, and we illustrate its application on gene expression and drug review data sets.