Statistical Properties of Lasso

Junwei Lu
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Junwei Lu: Harvard University

Chapter Chapter 10 in Big Data Analysis, 2025, pp 59-64 from Springer

Abstract: Abstract In this chapter, we return to the noisy linear regression. Recall the sparse linear model Y = 𝕏 β ∗ + ε $$Y = \mathbb {X} \beta ^* + \varepsilon $$ , where 𝕏 ∈ ℝ n × d $$\mathbb {X} \in \mathbb {R}^{n\times d}$$ and ∥ β ∗ ∥ 0 ≤ s $$\|\beta ^*\|_0 \le s$$ . We estimate the high-dimensional linear model via the Lasso estimator β ^ = arg min β 1 2 n ∥ Y − 𝕏 β ∥ 2 2 + λ ∥ β ∥ 1 . $$\displaystyle \widehat \beta = \operatorname *{\text{arg min}}_{\beta}\frac {1}{2n} \|Y-\mathbb {X}\beta \|_2^2 + \lambda \|\beta \|_1. $$ In this chapter, we will study the statistical properties of the Lasso estimator. Like the RIP condition for the basis pursuit, we also need conditions for Lasso.

Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-03161-7_10

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DOI: 10.1007/978-3-032-03161-7_10

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