THE DANTZIG SELECTOR: STATISTICAL ESTIMATION WHEN p IS MUCH LARGER THAN n^1

THE DANTZIG SELECTOR: STATISTICAL ESTIMATION WHEN p IS MUCH LARGER THAN n^1

2007, Vol. 35, No. 6, 2313-2351 | BY EMMANUEL CANDES^2 AND TERENCE TAO^3
The paper introduces the Dantzig selector, a new estimator for estimating a parameter vector $\beta$ in a linear model $y = X\beta + z$, where $y$ is a vector of observations, $X$ is a data matrix, and $z$ is a vector of stochastic measurement errors. The key assumption is that the number of variables $p$ is much larger than the number of observations $n$. The Dantzig selector is a solution to an $\ell_1$-regularization problem with a constraint on the residual vector, designed to handle noisy data. The paper shows that if the matrix $X$ satisfies a uniform uncertainty principle and the true parameter vector $\beta$ is sufficiently sparse, the Dantzig selector can estimate $\beta$ with high probability, achieving a mean squared error that is within a logarithmic factor of the ideal mean squared error one would achieve with perfect information about the nonzero coordinates of $\beta$. The results are nonasymptotic and provide explicit constants. The Dantzig selector is also shown to be robust against measurement errors and can be used for model selection, making it a versatile tool for handling high-dimensional data.The paper introduces the Dantzig selector, a new estimator for estimating a parameter vector $\beta$ in a linear model $y = X\beta + z$, where $y$ is a vector of observations, $X$ is a data matrix, and $z$ is a vector of stochastic measurement errors. The key assumption is that the number of variables $p$ is much larger than the number of observations $n$. The Dantzig selector is a solution to an $\ell_1$-regularization problem with a constraint on the residual vector, designed to handle noisy data. The paper shows that if the matrix $X$ satisfies a uniform uncertainty principle and the true parameter vector $\beta$ is sufficiently sparse, the Dantzig selector can estimate $\beta$ with high probability, achieving a mean squared error that is within a logarithmic factor of the ideal mean squared error one would achieve with perfect information about the nonzero coordinates of $\beta$. The results are nonasymptotic and provide explicit constants. The Dantzig selector is also shown to be robust against measurement errors and can be used for model selection, making it a versatile tool for handling high-dimensional data.
Reach us at info@study.space