# Sparse Signal Detection in Heteroscedastic Gaussian Sequence Models: Sharp Minimax Rates

arXiv: https://arxiv.org/abs/2211.08580 · HTML: https://mlsanity.com/paper/2211.08580/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 72 references that are in the map.
- First posted on arXiv 15 Nov 2022.

## Builds on (most cited first, 10 of 10)

- [Hypothesis Testing For Densities and High-Dimensional Multinomials: Sharp Local Minimax Rates](https://mlsanity.com/paper/1706.10003/) (2017)
- [Locally private non-asymptotic testing of discrete distributions is faster using interactive mechanisms](https://mlsanity.com/paper/2005.12601/) (2020)
- [Minimax rates in sparse, high-dimensional changepoint detection](https://mlsanity.com/paper/1907.10012/) (2019)
- [ℓp Testing and Learning of Discrete Distributions](https://mlsanity.com/paper/1412.2314/) (2014)
- [Power-enhanced simultaneous test of high-dimensional mean vectors and covariance matrices with application to gene-set testing](https://mlsanity.com/paper/2109.15287/) (2021)
- [Higher Criticism for Discriminating Word-Frequency Tables and Testing Authorship](https://mlsanity.com/paper/1911.01208/) (2019)
- [Minimax rates for sparse signal detection under correlation](https://mlsanity.com/paper/2110.12966/) (2021)
- [Simulation-Based Hypothesis Testing of High Dimensional Means Under Covariance Heterogeneity](https://mlsanity.com/paper/1406.1939/) (2014)
- [Local minimax rates for closeness testing of discrete distributions](https://mlsanity.com/paper/1902.01219/) (2019)
- [Distribution and correlation free two-sample test of high-dimensional means](https://mlsanity.com/paper/1904.07416/) (2019)

## Built on by (most cited first, 3 of 3)

- [Optimal community detection in dense bipartite graphs](https://mlsanity.com/paper/2505.18372/) (2025)
- [Minimax Signal Detection in Sparse Additive Models](https://mlsanity.com/paper/2304.09398/) (2023)
- [Bivariate Matrix-valued Linear Regression (BMLR): Finite-sample performance under Identifiability and Sparsity Assumptions](https://mlsanity.com/paper/2412.17749/) (2024)

## Abstract (arXiv)

Given a heterogeneous Gaussian sequence model with unknown mean θ ∈ mathbb Rd and known covariance matrix Sigma = operatornamediag(σ12,dots, σd2), we study the signal detection problem against sparse alternatives, for known sparsity s. Namely, we characterize how large ε^*>0 should be, in order to distinguish with high probability the null hypothesis θ=0 from the alternative composed of s-sparse vectors in mathbb Rd, separated from 0 in Lt norm (t ∈ [1,∞]) by at least ε^*. We find minimax upper and lower bounds over the minimax separation radius ε^* and prove that they are always matching. We also derive the corresponding minimax tests achieving these bounds. Our results reveal new phase transitions regarding the behavior of ε^* with respect to the level of sparsity, to the Lt metric, and to the heteroscedasticity profile of Sigma. In the case of the Euclidean (i.e. L2) separation, we bridge the remaining gaps in the literature.