# EigenPrism: Inference for High-Dimensional Signal-to-Noise Ratios

arXiv: https://arxiv.org/abs/1505.02097 · HTML: https://mlsanity.com/paper/1505.02097/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 May 2015.

## Built on by (most cited first, 4 of 4)

- [Statistical-Computational Tradeoffs in Mixed Sparse Linear Regression](https://mlsanity.com/paper/2303.02118/) (2023)
- [Noise Covariance Estimation in Multi-Task High-dimensional Linear Models](https://mlsanity.com/paper/2206.07256/) (2022)
- [Bayes optimal learning in high-dimensional linear regression with network side information](https://mlsanity.com/paper/2306.05679/) (2023)
- [FaCT: Faithful Concept Traces for Explaining Neural Network Decisions](https://mlsanity.com/paper/2510.25512/) (2025)

## Abstract (arXiv)

Consider the following three important problems in statistical inference, namely, constructing confidence intervals for (1) the error of a high-dimensional (p>n) regression estimator, (2) the linear regression noise level, and (3) the genetic signal-to-noise ratio of a continuous-valued trait (related to the heritability). All three problems turn out to be closely related to the little-studied problem of performing inference on the ℓ2-norm of the signal in high-dimensional linear regression. We derive a novel procedure for this, which is asymptotically correct when the covariates are multivariate Gaussian and produces valid confidence intervals in finite samples as well. The procedure, called EigenPrism, is computationally fast and makes no assumptions on coefficient sparsity or knowledge of the noise level. We investigate the width of the EigenPrism confidence intervals, including a comparison with a Bayesian setting in which our interval is just 5% wider than the Bayes credible interval. We are then able to unify the three aforementioned problems by showing that the EigenPrism procedure with only minor modifications is able to make important contributions to all three. We also investigate the robustness of coverage and find that the method applies in practice and in finite samples much more widely than just the case of multivariate Gaussian covariates. Finally, we apply EigenPrism to a genetic dataset to estimate the genetic signal-to-noise ratio for a number of continuous phenotypes.