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Diffstat (limited to 'paper')
| -rw-r--r-- | paper/sections/results.tex | 13 |
1 files changed, 8 insertions, 5 deletions
diff --git a/paper/sections/results.tex b/paper/sections/results.tex index 5d63cbd..1688aae 100644 --- a/paper/sections/results.tex +++ b/paper/sections/results.tex @@ -241,18 +241,21 @@ which is also a consequence of Theorem 1 in \cite{Negahban:2009}. \begin{theorem} \label{thm:approx_sparse} -Suppose the relaxed {\bf(RE)} assumption holds for the Hessian $\nabla^2 -f(\theta^*)$ and for the following set: +Suppose the {\bf(RE)} assumption holds for the Hessian $\nabla^2 +f(\theta^*)$ and $\tau_{\mathcal{L}}(\theta^*) = \frac{\kappa_2\log m}{n}\|\theta^*\|_1$ +on the following set: \begin{align} \nonumber -{\cal C}' \defeq & \{X \in \mathbb{R}^p : \|X_{S^c}\|_1 \leq 3 \|X_S\|_1 + 4 \|\theta^*_{\lfloor s \rfloor}\|_1 \} \\ \nonumber +{\cal C}' \defeq & \{X \in \mathbb{R}^p : \|X_{S^c}\|_1 \leq 3 \|X_S\|_1 ++ 4 \|\theta^* - \theta^*_{\lfloor s \rfloor}\|_1 \} \\ \nonumber & \cap \{ \|X\|_1 \leq 1 \} \end{align} -By solving \eqref{eq:pre-mle} for $\lambda \defeq 2\sqrt{\frac{\log m}{\alpha n^{1 - \delta}}}$ we have: +If the number of measurements $n\geq \frac{64\kappa_2}{\gamma}s\log m$, then +by solving \eqref{eq:pre-mle} for $\lambda \defeq 2\sqrt{\frac{\log m}{\alpha n^{1 - \delta}}}$ we have: \begin{align*} \|\hat \theta - \theta^* \|_2 \leq \frac{3}{\gamma} \sqrt{\frac{s\log m}{\alpha n^{1-\delta}}} - + \frac{2}{\gamma} \sqrt[4]{\frac{s\log m}{\alpha n^{1-\delta}}} \|\theta^* - \theta^*_{\lfloor s \rfloor}\|_1 + + 4 \sqrt[4]{\frac{s\log m}{\gamma^4\alpha n^{1-\delta}}} \|\theta^* - \theta^*_{\lfloor s \rfloor}\|_1 \end{align*} \end{theorem} |