# Spatially relaxed inference on high-dimensional linear models

arXiv: https://arxiv.org/abs/2106.02590 · HTML: https://mlsanity.com/paper/2106.02590/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 61 references that are in the map.
- First posted on arXiv 4 Jun 2021.

## Builds on (most cited first, 22 of 22)

- [Controlling the false discovery rate via knockoffs](https://mlsanity.com/paper/1404.5609/) (2014)
- [Exact post-selection inference, with application to the lasso](https://mlsanity.com/paper/1311.6238/) (2013)
- [Confidence Intervals and Hypothesis Testing for High-Dimensional Regression](https://mlsanity.com/paper/1306.3171/) (2013)
- [Exact Post-Selection Inference for Sequential Regression Procedures](https://mlsanity.com/paper/1401.3889/) (2014)
- [The Lasso with general Gaussian designs with applications to hypothesis testing](https://mlsanity.com/paper/2007.13716/) (2020)
- [De-biasing the Lasso: Optimal Sample Size for Gaussian Designs](https://mlsanity.com/paper/1508.02757/) (2015)
- [A General Theory of Hypothesis Tests and Confidence Regions for Sparse High Dimensional Models](https://mlsanity.com/paper/1412.8765/) (2014)
- [A significance test for the lasso](https://mlsanity.com/paper/1301.7161/) (2013)
- [Fast and Powerful Conditional Randomization Testing via Distillation](https://mlsanity.com/paper/2006.03980/) (2020)
- [High-Dimensional Inference: Confidence Intervals, p-Values and R-Software hdi](https://mlsanity.com/paper/1408.4026/) (2014)
- [De-Biasing The Lasso With Degrees-of-Freedom Adjustment](https://mlsanity.com/paper/1902.08885/) (2019)
- [A Study of Error Variance Estimation in Lasso Regression](https://mlsanity.com/paper/1311.5274/) (2013)
- [High-dimensional simultaneous inference with the bootstrap](https://mlsanity.com/paper/1606.03940/) (2016)
- [Estimating the error variance in a high-dimensional linear model](https://mlsanity.com/paper/1712.02412/) (2017)
- [Efficient Smoothed Concomitant Lasso Estimation for High Dimensional Regression](https://mlsanity.com/paper/1606.02702/) (2016)
- [Aggregation of Multiple Knockoffs](https://mlsanity.com/paper/2002.09269/) (2020)
- [Statistical control for spatio-temporal MEG/EEG source imaging with desparsified multi-task Lasso](https://mlsanity.com/paper/2009.14310/) (2020)
- [The Benefit of Group Sparsity in Group Inference with De-biased Scaled Group Lasso](https://mlsanity.com/paper/1412.4170/) (2014)
- [Group-bound: confidence intervals for groups of variables in sparse high-dimensional regression without assumptions on the design](https://mlsanity.com/paper/1309.3489/) (2013)
- [Familywise Error Rate Control via Knockoffs](https://mlsanity.com/paper/1505.06549/) (2015)
- [Discovering Conditionally Salient Features with Statistical Guarantees](https://mlsanity.com/paper/1905.12177/) (2019)
- [Recursive nearest agglomeration (ReNA): fast clustering for approximation of structured signals](https://mlsanity.com/paper/1609.04608/) (2016)

## Built on by (most cited first, 1 of 1)

- [A Conditional Randomization Test for Sparse Logistic Regression in High-Dimension](https://mlsanity.com/paper/2205.14613/) (2022)

## Abstract (arXiv)

We consider the inference problem for high-dimensional linear models, when covariates have an underlying spatial organization reflected in their correlation. A typical example of such a setting is high-resolution imaging, in which neighboring pixels are usually very similar. Accurate point and confidence intervals estimation is not possible in this context with many more covariates than samples, furthermore with high correlation between covariates. This calls for a reformulation of the statistical inference problem, that takes into account the underlying spatial structure: if covariates are locally correlated, it is acceptable to detect them up to a given spatial uncertainty. We thus propose to rely on the δ-FWER, that is the probability of making a false discovery at a distance greater than δ from any true positive. With this target measure in mind, we study the properties of ensembled clustered inference algorithms which combine three techniques: spatially constrained clustering, statistical inference, and ensembling to aggregate several clustered inference solutions. We show that ensembled clustered inference algorithms control the δ-FWER under standard assumptions for δ equal to the largest cluster diameter. We complement the theoretical analysis with empirical results, demonstrating accurate δ-FWER control and decent power achieved by such inference algorithms.