Abstract
| Original language | English |
|---|---|
| Pages (from-to) | 741-766 |
| Journal | Electronic Journal of Statistics |
| Volume | 12 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 27 Feb 2018 |
| Externally published | Yes |
Keywords
- Sparse linear regression
- minimax rates
- high-dimensional statistics
- adaptivity
- square-root estimators
Cite this
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Improved bounds for Square-Root Lasso and Square-Root Slope. / Derumigny, Alexis.
In: Electronic Journal of Statistics, Vol. 12, No. 1, 27.02.2018, p. 741-766.Research output: Contribution to journal › Article › Academic › peer-review
TY - JOUR
T1 - Improved bounds for Square-Root Lasso and Square-Root Slope
AU - Derumigny, Alexis
PY - 2018/2/27
Y1 - 2018/2/27
N2 - Extending the results of Bellec, Lecué and Tsybakov [1] to the setting of sparse high-dimensional linear regression with unknown variance, we show that two estimators, the Square-Root Lasso and the Square-Root Slope can achieve the optimal minimax prediction rate, which is (s/n)log(p/s), up to some constant, under some mild conditions on the design matrix. Here, n is the sample size, p is the dimension and s is the sparsity parameter. We also prove optimality for the estimation error in the lq-norm, with q∈[1,2] for the Square-Root Lasso, and in the l2 and sorted l1 norms for the Square-Root Slope. Both estimators are adaptive to the unknown variance of the noise. The Square-Root Slope is also adaptive to the sparsity s of the true parameter. Next, we prove that any estimator depending on s which attains the minimax rate admits an adaptive to s version still attaining the same rate. We apply this result to the Square-root Lasso. Moreover, for both estimators, we obtain valid rates for a wide range of confidence levels, and improved concentration properties as in [1] where the case of known variance is treated. Our results are non-asymptotic.
AB - Extending the results of Bellec, Lecué and Tsybakov [1] to the setting of sparse high-dimensional linear regression with unknown variance, we show that two estimators, the Square-Root Lasso and the Square-Root Slope can achieve the optimal minimax prediction rate, which is (s/n)log(p/s), up to some constant, under some mild conditions on the design matrix. Here, n is the sample size, p is the dimension and s is the sparsity parameter. We also prove optimality for the estimation error in the lq-norm, with q∈[1,2] for the Square-Root Lasso, and in the l2 and sorted l1 norms for the Square-Root Slope. Both estimators are adaptive to the unknown variance of the noise. The Square-Root Slope is also adaptive to the sparsity s of the true parameter. Next, we prove that any estimator depending on s which attains the minimax rate admits an adaptive to s version still attaining the same rate. We apply this result to the Square-root Lasso. Moreover, for both estimators, we obtain valid rates for a wide range of confidence levels, and improved concentration properties as in [1] where the case of known variance is treated. Our results are non-asymptotic.
KW - Sparse linear regression
KW - minimax rates
KW - high-dimensional statistics
KW - adaptivity
KW - square-root estimators
U2 - 10.1214/18-EJS1410
DO - 10.1214/18-EJS1410
M3 - Article
VL - 12
SP - 741
EP - 766
JO - Electronic Journal of Statistics
JF - Electronic Journal of Statistics
SN - 1935-7524
IS - 1
ER -