# The Sample Complexity of Robust Covariance Testing

arXiv: https://arxiv.org/abs/2012.15802 · HTML: https://mlsanity.com/paper/2012.15802/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Dec 2020.

## Built on by (most cited first, 5 of 5)

- [Privacy Induces Robustness: Information-Computation Gaps and Sparse Mean Estimation](https://mlsanity.com/paper/2211.00724/) (2022)
- [Private High-Dimensional Hypothesis Testing](https://mlsanity.com/paper/2203.01537/) (2022)
- [Robust Kernel Hypothesis Testing under Data Corruption](https://mlsanity.com/paper/2405.19912/) (2024)
- [Robust Testing and Estimation under Manipulation Attacks](https://mlsanity.com/paper/2104.10740/) (2021)
- [Robust Testing in High-Dimensional Sparse Models](https://mlsanity.com/paper/2205.07488/) (2022)

## Abstract (arXiv)

We study the problem of testing the covariance matrix of a high-dimensional Gaussian in a robust setting, where the input distribution has been corrupted in Huber's contamination model. Specifically, we are given i.i.d. samples from a distribution of the form Z = (1-ε) X + ε B, where X is a zero-mean and unknown covariance Gaussian N(0, Sigma), B is a fixed but unknown noise distribution, and ε>0 is an arbitrarily small constant representing the proportion of contamination. We want to distinguish between the cases that Sigma is the identity matrix versus γ-far from the identity in Frobenius norm. In the absence of contamination, prior work gave a simple tester for this hypothesis testing task that uses O(d) samples. Moreover, this sample upper bound was shown to be best possible, within constant factors. Our main result is that the sample complexity of covariance testing dramatically increases in the contaminated setting. In particular, we prove a sample complexity lower bound of Omega(d2) for ε an arbitrarily small constant and γ = 1/2. This lower bound is best possible, as O(d2) samples suffice to even robustly em learn the covariance. The conceptual implication of our result is that, for the natural setting we consider, robust hypothesis testing is at least as hard as robust estimation.