# Nonasymptotic one-and two-sample tests in high dimension with unknown covariance structure

arXiv: https://arxiv.org/abs/2109.01730 · HTML: https://mlsanity.com/paper/2109.01730/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 45 references that are in the map.
- First posted on arXiv 1 Sep 2021.

## Builds on (most cited first, 8 of 8)

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [A Kernel Test of Goodness of Fit](https://mlsanity.com/paper/1602.02964/) (2016)
- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [On the Optimality of Kernel-Embedding Based Goodness-of-Fit Tests](https://mlsanity.com/paper/1709.08148/) (2017)
- [High-Dimensional Multi-Task Averaging and Application to Kernel Mean Embedding](https://mlsanity.com/paper/2011.06794/) (2020)
- [Testing relevant hypotheses in functional time series via self-normalization](https://mlsanity.com/paper/1809.06092/) (2018)
- [Local minimax rates for closeness testing of discrete distributions](https://mlsanity.com/paper/1902.01219/) (2019)
- [Functional data analysis in the Banach space of continuous functions](https://mlsanity.com/paper/1710.07781/) (2017)

## Built on by (most cited first, 2 of 2)

- [A kernel-based analysis of Laplacian Eigenmaps](https://mlsanity.com/paper/2402.16481/) (2024)
- [Adaptive Personalized Federated Learning via Multi-task Averaging of Kernel Mean Embeddings](https://mlsanity.com/paper/2603.02233/) (2026)

## Abstract (arXiv)

Let X = (Xi)1≤ i ≤ n be an i.i.d. sample of square-integrable variables in ℝd, GBwith common expectation μ and covariance matrix Sigma, both unknown. We consider the problem of testing if μ is η-close to zero, i.e. |μ| ≤ η  against |μ| ≥ (η + δ); we also tackle the more general two-sample mean closeness (also known as em relevant difference) testing problem. The aim of this paper is to obtain nonasymptotic upper and lower bounds on the minimal separation distance δ such that we can control both the Type I and Type II errors at a given level. The main technical tools are concentration inequalities, first for a suitable estimator of |μ|2 used a test statistic, and secondly for estimating the operator and Frobenius norms of Sigma coming into the quantiles of said test statistic. These properties are obtained for Gaussian and bounded distributions. A particular attention is given to the dependence in the pseudo-dimension d_* of the distribution, defined as d_* := |Sigma|22/|Sigma|_∞2. In particular, for η=0, the minimum separation distance is Theta( d_*^frac14√|Sigma|_∞/n), in contrast with the minimax estimation distance for μ, which is Theta(de^frac12√|Sigma|_∞/n) (where de:=|Sigma|1/|Sigma|_∞). This generalizes a phenomenon spelled out in particular by Baraud (2002).