# Chi-square and normal inference in high-dimensional multi-task regression

arXiv: https://arxiv.org/abs/2107.07828 · HTML: https://mlsanity.com/paper/2107.07828/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 55 references that are in the map.
- First posted on arXiv 16 Jul 2021.

## Builds on (most cited first, 15 of 15)

- [Confidence Intervals and Hypothesis Testing for High-Dimensional Regression](https://mlsanity.com/paper/1306.3171/) (2013)
- [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)
- [Hypothesis Testing in High-Dimensional Regression under the Gaussian Random Design Model: Asymptotic Theory](https://mlsanity.com/paper/1301.4240/) (2013)
- [Group SLOPE - adaptive selection of groups of predictors](https://mlsanity.com/paper/1610.04960/) (2016)
- [De-Biasing The Lasso With Degrees-of-Freedom Adjustment](https://mlsanity.com/paper/1902.08885/) (2019)
- [Linear Hypothesis Testing in Dense High-Dimensional Linear Models](https://mlsanity.com/paper/1610.02987/) (2016)
- [Statistical control for spatio-temporal MEG/EEG source imaging with desparsified multi-task Lasso](https://mlsanity.com/paper/2009.14310/) (2020)
- [Testability of high-dimensional linear models with non-sparse structures](https://mlsanity.com/paper/1802.09117/) (2018)
- [Group Inference in High Dimensions with Applications to Hierarchical Testing](https://mlsanity.com/paper/1909.01503/) (2019)
- [The Benefit of Group Sparsity in Group Inference with De-biased Scaled Group Lasso](https://mlsanity.com/paper/1412.4170/) (2014)
- [Optimal Statistical Inference for Individualized Treatment Effects in High-dimensional Models](https://mlsanity.com/paper/1904.12891/) (2019)
- [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)
- [Asymptotic Confidence Regions for High-dimensional Structured Sparsity](https://mlsanity.com/paper/1706.09231/) (2017)
- [Support recovery and sup-norm convergence rates for sparse pivotal estimation](https://mlsanity.com/paper/2001.05401/) (2020)

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

- [Corrected generalized cross-validation for finite ensembles of penalized estimators](https://mlsanity.com/paper/2310.01374/) (2023)
- [Noise Covariance Estimation in Multi-Task High-dimensional Linear Models](https://mlsanity.com/paper/2206.07256/) (2022)

## Abstract (arXiv)

The paper proposes chi-square and normal inference methodologies for the unknown coefficient matrix B^* of size p× T in a Multi-Task (MT) linear model with p covariates, T tasks and n observations under a row-sparse assumption on B^*. The row-sparsity s, dimension p and number of tasks T are allowed to grow with n. In the high-dimensional regime pggg n, in order to leverage row-sparsity, the MT Lasso is considered. We build upon the MT Lasso with a de-biasing scheme to correct for the bias induced by the penalty. This scheme requires the introduction of a new data-driven object, coined the interaction matrix, that captures effective correlations between noise vector and residuals on different tasks. This matrix is psd, of size T× T and can be computed efficiently. The interaction matrix lets us derive asymptotic normal and chi2T results under Gaussian design and fracsT+slog(p/s)n→0 which corresponds to consistency in Frobenius norm. These asymptotic distribution results yield valid confidence intervals for single entries of B^* and valid confidence ellipsoids for single rows of B^*, for both known and unknown design covariance Sigma. While previous proposals in grouped-variables regression require row-sparsity slesssim√ n up to constants depending on T and logarithmic factors in n,p, the de-biasing scheme using the interaction matrix provides confidence intervals and chi2T confidence ellipsoids under the conditions min(T2,log8p)/n→ 0 and  fracsT+slog(p/s)+|Sigma-1ej|0log pn→0, quad fracmin(s,|Sigma-1ej|0)√ n √[T+log(p/s)]log p→ 0,  allowing row-sparsity sggg√ n when |Sigma-1ej|0 √ Tlll √n up to logarithmic factors.