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In this exercise, we work through a proof of (a slightly weaker version of) Proposition 10.2. (a) For any subset $S$ of cardinality $k$, the set $\mathbb{C}(S) \cap \mathbb{B}_2(1)$ is contained within the set $\mathbb{L}_1(r)$ with $r=4 k$. (b) Now show that if $\operatorname{RIP}(8 k, \delta)$ holds with $\delta<1 / 4$, then the restricted nullspace property holds. (Hint: Part (b) of Exercise 10.7 could be useful.)

   In this exercise, we work through a proof of (a slightly weaker version of) Proposition 10.2.
(a) For any subset $S$ of cardinality $k$, the set $\mathbb{C}(S) \cap \mathbb{B}_2(1)$ is contained within the set $\mathbb{L}_1(r)$ with $r=4 k$.
(b) Now show that if $\operatorname{RIP}(8 k, \delta)$ holds with $\delta<1 / 4$, then the restricted nullspace property holds. (Hint: Part (b) of Exercise 10.7 could be useful.)

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Statistical Learning with Sparsity: The Lasso and Generalizations

Statistical Learning with Sparsity: The Lasso and Generalizations

Trevor Hastie,… 1st Edition

Chapter 10, Problem 8

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In this exercise, we work through a proof of (a slightly weaker version of) Proposition 10.2. (a) For any subset $S$ of cardinality $k$, the set $\mathbb{C}(S) \cap \mathbb{B}_2(1)$ is contained within the set $\mathbb{L}_1(r)$ with $r=4 k$. (b) Now show that if $\operatorname{RIP}(8 k, \delta)$ holds with $\delta<1 / 4$, then the restricted nullspace property holds. (Hint: Part (b) of Exercise 10.7 could be useful.)

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