# Sparse model selection

September 5, 2016 — October 2, 2020

estimator distribution

functional analysis

high d

linear algebra

model selection

probability

signal processing

sparser than thou

statistics

On choosing the right model and regularisation parameter in sparse regression, which turn out to be nearly the same, and closely coupled to doing the regression. There are some wrinkles.

🏗 Talk about when degrees-of-freedom penalties work, when cross-validation and so on.

## 1 FOCI

The new hotness sweeping the world is FOCI, a sparse model selection procedure (Azadkia and Chatterjee 2019) based on Chatterjee’s ξ statistic as an independence test test. (Chatterjee 2020). Looks interesting.

## 2 Stability selection

🏗

For now see mplot for an introduction.

## 3 Relaxed Lasso

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## 4 Dantzig Selector

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## 5 Garotte

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## 6 Degrees-of-freedom penalties

See degrees of freedom.

## 7 References

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*The Annals of Statistics*.
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*arXiv:1910.12327 [Cs, Math, Stat]*.
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*Journal of Machine Learning Research*.
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*arXiv:1511.01650 [Cs, Math]*.
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*The Annals of Statistics*.
Bayati, and Montanari. 2012. “The LASSO Risk for Gaussian Matrices.”

*IEEE Transactions on Information Theory*.
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*The Annals of Statistics*.
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*Annales de l’Institut Henri Poincaré, Probabilités Et Statistiques*.
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*The Annals of Statistics*.
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*arXiv:1111.1162 [Cs, Math, Stat]*.
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*University of California, Berkeley*.
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*International Statistical Review*.
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*Bernoulli*.
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*Computational Statistics & Data Analysis*.
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*Bernoulli*.
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*Statistical Learning with Sparsity: The Lasso and Generalizations*.
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Hirose, Tateishi, and Konishi. 2011. “Efficient Algorithm to Select Tuning Parameters in Sparse Regression Modeling with Regularization.”

*arXiv:1109.2411 [Stat]*.
Huang, Cheang, and Barron. 2008. “Risk of Penalized Least Squares, Greedy Selection and L1 Penalization for Flexible Function Libraries.”

Janková, and van de Geer. 2016. “Confidence Regions for High-Dimensional Generalized Linear Models Under Sparsity.”

*arXiv:1610.01353 [Math, Stat]*.
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*Journal of Machine Learning Research*.
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*Journal of Multivariate Analysis*.
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*Journal of Machine Learning Research*.
Koltchinskii. 2011.

*Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems*. Lecture Notes in Mathematics École d’Été de Probabilités de Saint-Flour 2033.
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*Annals of Statistics*.
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*arXiv:2004.11554 [Stat]*.
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*arXiv:1311.6238 [Math, Stat]*.
Lemhadri, Ruan, Abraham, et al. 2021. “LassoNet: A Neural Network with Feature Sparsity.”

*Journal of Machine Learning Research*.
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*Journal of Statistical Planning and Inference*.
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*arXiv:1609.07195 [Stat]*.
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*The Annals of Statistics*.
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*Advances in Neural Information Processing Systems*.
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*The Annals of Statistics*.
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*The Annals of Statistics*.
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*Biometrika*.
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*The Annals of Statistics*.
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*Statistical Science*.
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*Probability Theory and Related Fields*.
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*The Annals of Statistics*.
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*arXiv:1908.01755 [Cs, Stat]*.
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*Journal of the American Statistical Association*.
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*Journal of the American Statistical Association*.
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*The Annals of Statistics*.
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*Electronic Journal of Statistics*.
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*Journal of Business & Economic Statistics*.
Xu, Caramanis, and Mannor. 2012. “Sparse Algorithms Are Not Stable: A No-Free-Lunch Theorem.”

*IEEE Transactions on Pattern Analysis and Machine Intelligence*.
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*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.
———. 2007. “Model Selection and Estimation in the Gaussian Graphical Model.”

*Biometrika*.
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*The Annals of Statistics*.
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*Journal of the American Statistical Association*.
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*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.
Zhao, Rocha, and Yu. 2006. “Grouped and Hierarchical Model Selection Through Composite Absolute Penalties.”

———. 2009. “The Composite Absolute Penalties Family for Grouped and Hierarchical Variable Selection.”

*The Annals of Statistics*.
Zhao, and Yu. 2006. “On Model Selection Consistency of Lasso.”

*Journal of Machine Learning Research*.
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*Journal of the American Statistical Association*.
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*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.
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*The Annals of Statistics*.