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\label{EXO:sparse:weaklq} For any [math]q \gt 0[/math], a vector [math]\theta \in \R^d[/math] is said to be in a weak [math]\ell_q[/math] ball of radius [math]R[/math] if the decreasing rearrangement [math]|\theta_{[1]}|\ge |\theta_{[2]}| \ge \dots[/math] satisfies

[[math]] |\theta_{[j]}|\le Rj^{-1/q}\,. [[/math]]

Moreover, we define the weak [math]\ell_q[/math] norm of [math]\theta[/math] by

[[math]] |\theta|_{w\ell_q}=\max_{1\le j\le d} j^{1/q}|\theta_{[j]}|. [[/math]]

• Give examples of [math]\theta, \theta' \in \R^d[/math] such that
  [[math]] |\theta+\theta'|_{w\ell_1} \gt |\theta|_{w\ell_1}+|\theta'|_{w\ell_1} [[/math]]
  What do you conclude?
• Show that [math]|\theta|_{w\ell_q} \le |\theta|_{q}[/math].
• Given a sequence [math] \theta_1, \theta_2, \dots [/math], show that if [math]\lim_{d \to \infty}|\theta_{\{1, \dots, d\}}|_{w\ell_q} \lt \infty[/math], then [math]\lim_{d \to \infty}|\theta_{\{1, \dots, d\}}|_{q'} \lt \infty[/math] for all [math]q' \gt q[/math].
• Show that, for any [math]q \in (0,2)[/math] if [math]\lim_{d \to \infty}|\theta_{\{1, \dots, d\}}|_{w\ell_q}=C[/math], there exists a constant [math]C_q \gt 0[/math] that depends on [math]q[/math] but not on [math]d[/math] such that under the assumptions of Theorem~Linear Regression Model, it holds
  [[math]] |\thetahard-\theta^*|_2^2\le C_q\Big(\frac{\sigma^2 \log 2d}{n}\Big)^{1-\frac{q}{2}} [[/math]]
  with probability .99.