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Journal Articles IEEE Transactions on Information Theory Year : 2014

## Optimal Designs for Lasso and Dantzig Selector Using Expander Codes

Yohann de Castro

#### Abstract

We investigate the high-dimensional regression problem using adjacency matrices of unbalanced expander graphs. In this frame, we prove that the $\ell_{2}$-prediction error and the $\ell_{1}$-risk of the lasso and the Dantzig selector are optimal up to an explicit multiplicative constant. Thus we can estimate a high-dimensional target vector with an error term similar to the one obtained in a situation where one knows the support of the largest coordinates in advance. Moreover, we show that these design matrices have an explicit restricted eigenvalue. Precisely, they satisfy the restricted eigenvalue assumption and the compatibility condition with an explicit constant. Eventually, we capitalize on the recent construction of unbalanced expander graphs due to Guruswami, Umans, and Vadhan, to provide a deterministic polynomial time construction of these design matrices.

### Dates and versions

hal-01027523 , version 1 (22-07-2014)

### Identifiers

• HAL Id : hal-01027523 , version 1
• ARXIV :
• DOI :

### Cite

Yohann de Castro. Optimal Designs for Lasso and Dantzig Selector Using Expander Codes. IEEE Transactions on Information Theory, 2014, 60 (11), ⟨10.1109/TIT.2014.2353995⟩. ⟨hal-01027523⟩

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